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1

Preface

The purpose of this work is twofold: to provide a rigorous mathematical foundationfor study of the probability distributions of observables in quantum statistical mechanics,and to apply the theory to examples of physical interest. Although the work willprimarily interest mathematicians and mathematical physicists, I believe that results ofpurely physical interest (and at least one rather surprising result) are here as well. Indeed,some (§9.5) have been applied (see [JKS]) to study a model of the effect of angularmomentum on the frequency distribution of the cosmic background radiation. It issomewhat incongruous that in the half century since the development of quantumstatistics, the questions of probability distributions in so probabilistic a theory have beenaddressed so seldom. Credit is due to the Soviet mathematician Y.A. Khinchin, whoseMathematical Foundations of Quantum Statistics was the first comprehensive work (tomy knowledge) to address the subject.

Chapters 7 and 8 are a digression into probability theory whose physical applicationsappear in Chapter 9. These chapters may be read independently for their probabilisticcontent. I have tried wherever possible to make the functional analytic and operatortheoretic content independent of the probabilistic content, to make it accessible to a largergroup of mathematicians (and hopefully physicists).

My thanks go to I.E. Segal, whose ideas initiated this work and whose work hasprovided many of the results needed to draw up the framework developed here. Mythanks go also to Thomas Orowan, who saw the input and revision of this manuscript,using TEX, from beginning to end; his work was invariably fast and reliable. Finally Iwould like to express my appreciation to the Laboratory for Computer Science at M.I.T.,on whose DEC 10 computer this manuscript was compiled, revised, and edited.

2

Contents

Preface 1

Ch. 1 Introduction 4 1.1 Purposes and Background 4 1.2 The Free Boson and Fermion Fields Over a Hilbert Space 7 1.3 The State Probability Space 10

Ch. 2 Value Functions on a Canonical Ensemble 13 2.1 Physical Interpretation of Value Functions 13 2.2 Distribution of Occupation Numbers 14 2.3 Formalism of Asymptotic Spectral Densities 16

Ch. 3 Integrals of Independent Random Variables 20 3.1 General Theory of One Parameter Families 20 3.2 Background for Singular Integrals 26 3.3 Distributions Under Symmetric Statistics 27 3.4 Antisymmetric Distributions 32

Ch. 4 Singular Central Limit Theorems, I 34 4.1 The Case 34-²,³ ~ �, Á �²,³ � ��

4.2 The Case 38²�- ³ ²�³ � �Á �- ²�³ ~ �Z Z Z

Ch. 5 Singular Central Limit Theorems, II 47 5.1 Asymptotics Under Uniform Spectral Density 47 5.2 Asymptotics for Non-Vanishing Densities Near Zero 51

Ch. 6 Physical Applications 60 6.1 The Spectral Measure: An Example in Schrödinger Theory 60 6.2 The Spectral Approximation Theorem 61 6.3 Observable Distributions in Minkowski Space 62 6.4 Distributions in Einstein Space 69 6.5 Physical Discussion 74

Ch. 7 The Lebesgue Integral 76 7.1 Definitions and Probabilistic Background 77 7.2 Definition of the Lebesgue integral 83 7.3 Basic Properties 88

3

Ch. 8 Integrability Criteria and Some Applications 94 8.1 The Criterion 95 8.2 The Riemann Integral 106 8.3 The -Integral 1109i

Ch. 9 Joint Distributions and Applications 116 9.1 Integrals of Jointly Distributed R.V.'s 116 9.2 Abelian -algebras 117> i

9.3 Computations with Value Functions 119 9.4 Joint Asymptotic Distributions 121 9.5 Applications 122

Epilogue 126References 127Index 130

List of Symbols 13

4

Chapter 1

Introduction

§1.1 Purposes and Background

The most general mathematical description of an equilibrium quantum system is in itsdensity operator, which contains all information relevant to the probability distributionsof associated observables. Let be such a system, with Hamiltonian calculated in aI /reference frame with respect to which has zero mean linear and angular momentum.ILet be the inverse temperature of ( is temperature and is Boltzmann's� I~ ; ��

�;

constant). If the operator is trace class the appropriate density operator for is�c /� I

� ~ À�

�

c /

c /

�

�tr

Our general purpose is to obtain probability distributions of observables in fromIspectral properties of . If has very dense point spectrum, distributions depend on it/ /only through an appropriate spectral measure; if the spectrum has a continuouscomponent then has infinite trace, so that the associated density operator and�c /�

probability distributions must again be defined via spectral measures. We will study howdistributions are determined by spectral measures and apply the resulting theory to the�(continuous spectrum) invariant relativistic Hamiltonian in Minkowski space and to the(discrete spectrum) invariant Hamiltonian in a spherical geometry (Einstein space), toderive probability distributions of certain important observables.

In some systems with continuous spectrum, a natural obtains through a net � ¸/ ¹� ���

of operators with pure point spectrum which in an appropriate sense approximates . In/order that subsequent conclusions be well-founded, it must be required that be�independent of the choice of within some class of physically appropriate or¸/ ¹ Á� ���

“natural” nets. For example, if is an elliptic operator on a non-compact Riemannian(manifold , a natural class arises in approximating by large compact manifolds. The4 4well-developed theory of spectral asymptotics of pseudodifferential operators is usefulhere (see, e.g., [H], [See]). The procedure of infinite volume limits has also been studiedand developed in Schrödinger theory (see [Si, Section C6]). In physical applications suchas to the Planck law for photons, this approximation procedure is appropriate sincedistributions must be localized spatially, as well as with respect to wave propagationvector.

In the systems we consider, many interesting observables are sums of independentones indexed by the spectrum of a (maximal) commuting set of observables. Thus incases of continuous and asymptotically continuous spectrum, the notion of sum ofindependent random variables is very naturally replaced by that of an integral (over the

5

spectrum of observables), defined in a way completely analogous to the Riemann integral.The theory of such integrals and associated central limit theorems will be developed(Chapters 3-5), and then applied to particular random variables of interest.

A more comprehensive and abstract theory will be studied in Chapters 7-9. Theintegration procedure will be part of a more complete Lebesgue integration theory forrandom variable-valued functions. Connections will be made with the theory of randomdistributions and purely random fields [V,R1].

The related work in this area has been done primarily on , with Lebesgue measure.9�

General information on random distributions is contained in [GV]. Multi-dimensionalwhite noise was introduced in [Che], motivated by study of so-called Lévy Brownianmotion in [Lé]; the topic was further developed in [Mc]. A comprehensive theory ofgeneralized random fields was introduced by Malchan [M], using the notion ofbiorthogonality of random distributions on .9�

The non-linearity of the Lebesgue integral in Chapter 7 is not essential, since it isequivalent to an ordinary (linear) Lebesgue integral of a distribution-valued randomvariable field, and equivalently an integral over a space of logarithms of characteristicfunctions. Thus the integral is a simple and fundamental object. The probabilisticcontent is itself novel and (hopefully) interesting, and these chapters have been writtenlargely as a semi-autonomous part of the monograph. Probability and statisticalmechanics are re-joined in Chapter 9.

There are two physical implications of this work which warrant attention. The first isthat non-normally distributed observables (such as photon number) can arise in physicallyattainable situations, namely those in which spectral density is non-vanishing near zeroenergy; the latter occurs in systems which are approximately one-dimensional, forexample, in optic fibers or wave guides. The second is that within the class of models inwhich the density operator depends only on the Hamiltonian, the "blackbody spectrum'' ina large-scale equilibrium system of photons admits an energy density spectrum whichfollows a classical Planck law, whose specific form depends only on basic geometry.This work provides a rigorous basis for the study of blackbody radiation in general spatialgeometries, specifically the spherical geometry of Einstein space. This is a model of thephysical universe (see [Se2]), and is useful in mathematical physics, being a naturalspatially compact space-time which admits the action of the full conformal group.

Some of the results to be presented here are treated somewhat differently and in morespecific situations in an excellent foundational work by Khinchin [Kh], who treatsasymptotic distributions for photon systems in Minkowski space. The asymptoticspectrum of the Hamiltonian there is approximated to be concentrated on the integers;such a procedure suffices for consideration of energy observables for photons in threedimensional geometries, although it must be considered somewhat heuristic. It ishowever not adequate for treatment of more general situations (as will be seen here),since the very volatile dependence of distributions on spectral densities near 0 is notvisible in such an analysis. In general terms, however, the present work extends manyideas pioneered by Khinchin.

The reading of any chapter is perhaps best done in two stages, the first involving onlybrief inspections of technical aspects of proofs, and the second involving a more

6

thorough reading. Technicalities are often unavoidable in the proofs of theorems whosehypotheses are minimally technical.

We now provide a brief explanation of the structure of this monograph. Theremainder of this chapter provides a mathematical-physical framework. This includes adescription of Fock space, and a rigorous presentation of elementary results on quantumstatistical probability distributions. For example, we verify the common assumption thatoccupation numbers are independent random variables. Some of these results may befound (in somewhat less thorough form) in statistical physics texts. Chapter Twopresents some novel aspects of calculating distributions of observables (still in discretesituations), and presents fundamentals of the continuum limit. The basics of integrationtheory of random variable-valued functions are developed in Chapter 3. Chapters 4 and 5give applications to calculating observable distributions, including criteria for normalityand non-normality, and Chapter 6 provides physical applications. Here explicitdistributions are calculated, and a rigorous Planck law is derived for Bose and Fermiensembles. It is shown that the Planck laws are essentially independent of the intrinsicgeometry of large systems. Chapters 7 and 8 develop an analogous Lebesgue theory ofintegration, and Chapter 9 provides further applications, in the more general framework.

We dispose of a few technical preliminaries. Throughout this work (except in §9.5)we make the physical assumption that the chemical potential of particles underconsideration is 0; this does not involve essential loss of generality, as the general casecan be treated similarly. Probability distributions are studied for particles obeying Boseand Fermi statistics. The two situations are similar, and are studied in parallel. Theessentials of Segal's [Se3,4] formalism for free boson and fermion fields are usedthroughout and are described below.

We briefly mention some conventions. The symbols t h ] l, , and denote theintegers and the natural, complex, and real numbers, respectively. The non-negativeintegers and reals are represented by and . The functions with compactt lb b B*support are denoted by . The symbols , , and denote probability measure,*�

B F ; Lexpectation and variance. The point mass at 0 is represented by , while and �� ¬ ´ h µdenote convergence in law (or weak convergence) and the greatest integer function,respectively. The spectrum of an operator is ; the words “positive” and “non-( ²(³�negative” are used interchangeably, as are “increasing” and “non-decreasing”, etc.The abbreviations r.v., d.f., ch.f and a.s., mean “random variable,” “distributionfunction”, “characteristic function,” and “almost surely”. The notation w-lim denotes aweak limit, i.e., limit in distribution. denotes the normal distribution with mean 5²�Á �³ �and variance . If is an r.v. or a distribution, then denotes its d.f.� ? -?

We use the notation if is bounded as . We use�²%³ ~ 6²%³ ²% ¦ �³ % ¦ ��²%³

6²%³

�²%³ ~ �²%³ ²% ¦ �³ ¦ � % ¦ � if as .�²%³�²%³

A question may arise as to the dimensions of physical quantities. The system of unitswill be entirely general, unless otherwise specified. For instance, the energy and,inverse temperature may be interpreted in any units inverse to each other. Also, a�preference for setting will be evident in various places.� ~ �

7

§1.2 The Free Boson and Fermion Fields Over a Hilbert Space

We now present relevant aspects of free boson and fermion fields in their particlerepresentations. No proofs will be supplied; a more detailed and general description isgiven by Segal [Se3, Se4].

Let be a separable complex Hilbert space; for notational convenience and without>loss of generality assume is infinite dimensional. For let> h� �

A > > > >�

�~�

�

~ n nà n ~ ²�À�³�be the -fold tensor product of with itself, and be the unitary representation of� = ² h ³> �

²)³

the symmetric group of order on which is uniquely determined by the property' A� ��

= ² ³ % ~ % ² �  % � ³Â�²)³

�~� �~�

� �

� � �²�³� � ' >: ;� � �c�

= ² ³� �²)³ ²)³

� A thus permutes tensors. The (closed) subspace consisting of elements leftinvariant by is the D E= ² ³ ¢ � ��

²)³�� � ' -fold symmetrized tensor product of with>

itself.By convention A ]�

²)³~ À

For % Áà Á % �� � >

% v % và v % � % � = ² ³ % ²�À�³�

�[� � � � �

�~� �~�

� �

��²)³� � �: ;

� '�

�

is the orthogonal projection of into ; the latter is clearly spanned by vectorsn�~�

�

%� �²)³

A

of the form (1.2). Note that for and % � � Á� �> � '

� ��~� �~�

� �

� ²�³% ~ % À ²� 1.3)

The direct sum of symmetrized tensor products over all orders

A A)

�~�

B

�²)³� �

is the over .(Hilbert) space of symmetrized tensors >

Any unitary on can be lifted to a unitary which is< ²<³ ¢ ¦> ! A A� � �²)³ ²)³ ²)³

defined uniquely by

! >�²)³

�~� �~�

� �

� � �²< ³ % ~ ²<% ³ ²% � ³Â ²�À�³� �

8

! A� �²)³ ²)³

²< ³ � < is simply the restriction to of the -fold tensor product of with itself; weappend the convention that is the identity. We define! ] ]�

²:³²< ³ ¢ ¦

! !)

�~�

B

�²)³²< ³ ~ ²<³Á�

so that for % � Á� �A

! !) � �

�~� �~�

B B

�²)³²< ³ % ~ ²<³% À: ;� �

A self-adjoint operator in is naturally lifted to as the self-adjoint generator of the( > A)

one parameter unitary group :!)�!(²� ³

� ²(³ � ²� ³ Á ²�À³� �

! �!! !) )

²�!(³

!~�e

with the derivative taken in the strong operator topology.The definitions for antisymmetric statistics are fully analogous to those above. Let

² ³ � = ² h ³� � A denote the sign of , and be the unitary representation of on ' '� � ��²-³

defined by

= ² ³ % ~ ² ³ % ² � Á % � ³À�²-³

�~� �~�

� �

� � �²�³� � � ' >: ;� � �c�

Let denote the subspace of elements left invariant by , withA � �� �²-³ ²-³

�D E= ² ³ ¢ �'A ]�

²-³� ; the collection

% w % wà w % � % � = ² ³ % ²% � ³Â ²�À³�

�[� � � � � �

�~� �~�

� �

��²-³� � �: ;

� '�

� >

spans . The over isA >�²-³ space of antisymmetrized tensors

A A-

�~�

B

�²-³~ À ²�À�³�

If is unitary on , then is defined by< ²<³ ¢ ¦> ! A A� � �²-³ ²-³ ²-³

!�²-³

�~� �~�

� �

� �²< ³ % ~ <% Á: ;� �with the identity,! ] ]�

²-³¢ ¦

9

! !-

�~�

B

�²-³²< ³ � ²<³�

is the lifting of to . If is self-adjoint in , then is the generator of the< ( � ²(³A > !�²-³

-

unitary group .!-�!(²� ³

Henceforth statements not specifically referring to the (anti-)symmetric constructionswill hold under both statistics; in particular this will hold when subscripts and are) -omitted.

Definition 1.1: The operator is the of .� ²(³ (! quantization

The map acts linearly on bounded and unbounded self-adjoint operators to the�!extent that if are mutually orthogonal projections in and , then¸7 ¹ ¸, ¹ �� �� � ��h h> l

� , 7 ~ , � ²7 ³Â ²�À�³! !: ;� ��~� �~�

B B

� � � �

this fact will later prove useful. Physically, is the Hilbert space of states² Á � ²(³³A !together with the Hamiltonian of a many-particle non-interacting system each particle ofwhich has states in and time evolution governed by .> (

If is an orthonormal basis for then orthonormal bases for are. ~ ¸� ¹� �� �h > A

given by

8 h

8 h

�²)³

�~�

�

� � � � �

�²-³

�~�

�

� � � � �

~ � ¢ � � � � Ã � � Â � �

~ � ¢ � � � � Ã � � Â � �

J K�J K�

�

�

(symmetric)

(antisymmetric) .

²�À ³

The basis can be represented through the correspondence8�²)³

��~�

�

� � � �� © ²� Á� Áó ~ ²� � .³Á� �

n

where denotes the number of appearances of on the left hand side, and � � � ~ �À� � ��~�

B�If we append the convention , then) ~ ¸�¹ ��

²)³]

D A˜ ~ ) ���~�

B

�²)³

)

is an orthonormal basis for We can thus writeA)À

10

D t˜ ~ ²� Á� Áó ¢ � �  � � B Á ²�À���³J K�� � � �b

�~�

B

where corresponds to 1, the basis for D̃ A ]~ ²�Á �Áà ³ ~ À�²)³

Correspondingly, if

D AV ~ ) � Á��~�

B

�²-³

-

there is a bijection

��~�

�

� � �� © ²� Á� Áó ~�

n;

note that no above may be greater than 1, since 1 limits the number of appearances of��

�� under antisymmetric statistics (see (1.6)). The collection

DV ~ ²� Á� Áó ¢ � � ¸�Á �¹Â � � B ²�À���³J K�� � � �

�~�

B

is a basis of .A-

§1.3 The State Probability Space

Let be an orthonormal basis for , with the orthogonal projection. ~ ¸� ¹ 7� �� �h >

onto the span of . If is the basis of constructed in (1.10), and�� D A

n ~ ²� Á� Áó � Á� � D then

� ²7 ³ ~ � ²� � ³À ²�À��³! h� �n n

Consequently if and then by (1.8)¸� ¹ � * ~ � 7� �� � ��~�

B

h l �

� ²*³ ~ � � À! n n: ;��~�

B

� � (1.12)

Given a distinguished operator on such that is non-negative self-adjoint and� A �²�³ tr can be interpreted as a random variable on :²��³ ~ �Á � ²*³� ! D

Definition 1.2: An operator with properties and is a on .� A²�³ ²��³ density operator Through the probability measure ( , on , it forms the F � Dn n n³ ~ º » state probabilityspace value corresponding to . The r.v. (if defined) is the ² Á ³ �² ³ ~ º� ²*³ Á »D F � !n n nfunction of or of .� ²*³ *!

If is self adjoint on , , and is trace class,( � � �> � c � ²(³� !

11

� ~�

�

c � ²(³

c � ²(³

� !

� !tr

is a density operator. If diagonalizes and is as above, then the state. ~ ¸� ¹ (� ��h D

probability space corresponding to is the or ² Á ³D F � (symmetric antisymmetric)canonical ensemble over . Clearly any operator which commutes with defines an( * (r.v. in this way. These will be our object of study; of particular interest will be valuefunctions of number operators 5 ~ � ²7 ³À� �!

In physical applications is a positive Hamiltonian governing time evolution of(single particles in a multi-particle system. If , then represents a(� ~ , � � � .� � � �

physical single particle state of energy is the many-particle state,  ~ ²� Á� ó� � �Ánwith particles in state . By (1.11), the observable is the number� � ²� ~ �Á �Áà ³ 5� � �

of particles in state , and its value function is interpreted accordingly.��We now derive necessary and sufficient conditions for existence of a canonical

ensemble over , which we assume to be positive in . The following proposition has( >been implicit in the physical literature on statistical mechanics.

Definition 1.3: An operator is 0- 0 is outside its spectrum.free if

Proposition 1.4: The operator is trace class if and only if�c � ²(³� !

is trace class and is 0-free under symmetric statistics.²�³ � (c (�

is trace class under antisymmetric statistics.²��³ �c (�

Note that if either or is trace class, then both and have pure� � ( � ²(³c ( c � ²(³� � ! !point spectrum and finite multiplicities, since . Thus previous assumptions( ~ � ²(³O! A�

on have involved no loss of generality.(We sketch an argument for Proposition 1.4. In the symmetric case, define the

function on by . By (1.12)˜ n n n� � �D ² ³ ~ �

tr � ~ � À ²�À��³c � ²(³

�

c , ² ³� !

D

�) �~�

B

� ��n

n n

˜

�

On the other hand, when the product

� � �: ;�~� �~� �~�

B B Bc , c� c ,²� c � ³ ~ � Á ²�À��³� �� �

is multiplied out, it coincides with (1.13), so the latter converges if and only if

12

��~�

Bc ,� � BÁ� �

and no are 0. The proof in the antisymmetric case is similar.,�

We thus have

2 � � ~

2 ~ ²� c � ³

2 ~ ²� b � ³tr . (1.15)

(symmetric)

(antisymmetric)

c � ²(³

)�~�

Bc , c�

-�~�

Bc ,

� !

�

�

~��������

��

�

�

Corollary 1.4.1: If is trace class, then has pure point spectrum and finite� (c � ²(³� !

multiplicities.

13

Chapter 2

Value Functions on a Canonical Ensemble

Throughout this chapter is a positive self-adjoint (energy) operator in , satisfying( >²�³ ²��³ ( ¸, ¹ or of Proposition 1.4. In particular has pure point spectrum , and an� �~�

B

orthonormal eigenbasis , with . Assume is separable and (for. ~ ¸� ¹ (� ~ , �� �� � � �h >

notational convenience) infinite-dimensional, and define the number operator 5 ~ �� !

²7 ³ .� as before. The basis of corresponding to is given in (1.10); its genericD A

elements will be . Definen ~ ²� Á� Áó� �

� ~ À ²�À�³�

�

c � ²(³

c � ²(³

� !

� !tr

If is the value function of , and that of , then� � ²(³ � 5! � �

�²�³ ~ �², ³� À ²�À�³��~�

B

� �

More generally, if commutes with and has value function * ( *� ~ � � Á � ²*³� � ��~�

B

! �� �� �.

Definition 2.1: The r.v. is the in the canonical ensemble over� ��!� occupation number

(.

Note that in the ensemble, according to (1.12),

F �² ³ ~ º » � Á � À ²�À�³� �

� 2n n n n n , = =

tr c � ²(³

c , � c , �

� !

� �N O� ��~�

B

� � � �

§2.1 Physical Interpretation of Value Functions

The canonical ensemble over at inverse temperature describes a non-self-( � ��interacting system of particles in equilibrium at temperature ( is Boltzmann's; ~ ��

��

constant) with particles whose individual time evolution is governed by the Hamiltonian( *. If the operator , representing a physical observable for single particles, commuteswith , is the observable representing "total amount" of . Precisely,( � ²*³ *!

14

� ²*³ ~ � º*� Á � »! n n: ;��~�

B

� � � , (2.4)

º*� Á � » * �� � � representing the value of in the state .The value function of is a random variable on the canonical ensemble,� ²*³!

representing the probabilistic nature of . By its definition� ²*³!

�² ³ ~ � º*� Á � »À ²�À³n ��~�

B

� � �

In particular, the occupation number corresponding to is the r.v. on which� � ² Á ³� � D F

on takes the value , i.e., the number of particles in state . The value of n � � ~ , �� � � � �"

on is and represents the "total energy" of the particles in state .n , � Á �� � �

§2.2 Distribution of Occupation Numbers

If denote occupation numbers and if and� ~ ²! Á ! Áà ³Á ~ ²� Á � Áó� � � � �t nt nh ~ ! � ¸� ¹� � � � ��, then according to (2.3) the joint characteristic function of ish

) ;² ³ ~ ²� ~ � Á ²�À³�

2t � h � h c h

�

t n t n E n

n

) �D

�

where , and denotes mathematical expectation. The right side factorsE ~ ², Á, Áó� � ;to give

)² ³ ~�

2 � b � ³

²� c � ³t �H

�~�

B c , b�! c�

c , b�!

�

�

� �

� �

(symmetric)( (antisymmetric)

. (2.7)

Thus, occupation numbers are independent r.v.'s with characteristic functions��

)�

c , c , b�! c�

c , c� ²c , b�!³²!³ ~ ²�À�³

²� c � ³²� c � ³ À

²� b � ³ ²� b � ³H � �

� �

� �

� �

By inversion of (2.8), is geometric in the first case, with��

F²� ~ �³ ~ ²� c � ³� ²� ~ �Á �Áà ³Á ²�À ³�c , c , �� �� �

; L²� ³ ~ Á ²� ³ ~ Á ²�À��³� �

� c � ²� c �³� �, ,

,

�� �

�

� �

�

where denotes variance. Under antisymmetric statistics is Bernoulli, withL ��

15

F²� ~ �³ ~ ²�À��³²� b �³ � ~ �

²� b � ³  � ~ ��

, c�

c , c�H �

�

�

�

;

; L²� ³ ~ Á ²� ³ ~ À� �

� b � ²� b �³� �, ,

,

�� �

�

� �

�

The above distributions (specifically the independence of the occupation numbers)explicitly verify facts which have been used in the physical literature for some time.

Rewriting of (2.3) shows that if

D� � � �

�~�

B

~ ~ ²� Á� Áó ¢ O O � � ~ � Á ²�À��³J K� n n

the particle number value function satisfies� ~ ���~�

B

�

F t²� ~ �³ ~ � ~ � ²� � ³ ²�À��³� �

2 2� �n

E n

�

c h b

��� ���

c ,

D

��

� � �

�~�

�

��

�

under symmetric statistics, with replaced by in the antisymmetric case. There� �is a method of expressing (2.13) more simply for small values of which, however,�becomes more complicated as increases, and is not discussed here.�

The total energy

" ~ h ~ , �E n � � �

is a discrete r.v. with

7² ~ �³ ~ � ~ Á� � �²�³

2 2" �

n�

c �c �

D

��

�

where is the cardinality of : �²�³ ~ ¸ � h ~ �¹ÀD D� n n EWhen has uniformly spaced spectrum , (physically interesting in one-( , ~ �� �

dimensional situations), these specialize in the symmetric case to

16

F²� ~ �³ ~ ��

2

~ � ��

2

~ �� �

2 � c �

) ��� ��

c �

) ��� �� � ��

c �c �

) ��� ���

c � c �

�� �: ;�

� �

�~�

�

�

� �c� � �c�

�~�

�c�

��

� �c�

�~�

�c�

� �c�

��

����

�� ��

�

�

�c

) ��� ���

c � c� �

c c�

)

c� c� c� c� c�

�~�

� B

�~�b�

��

�� ���

�� ��

�� �� �� ��

~ �� �

2 ²� c � ³²� c � ³

~ à ~ � ²� c � ³ ~ � ²� c � ³À�

2

²�À��

�� �

� �c�

�~�

�c�

� �c��

�³

Similarly in the antisymmetric case

F²� ~ �³ ~ � ²� c � ³À ²�À���³�

2- �~�

�c �

c�²�b�³� � ��

Note that the tail of the number r.v. in (2.14) is geometric under symmetric statistics, andresembles a discretized Gaussian in the antisymmetric case.

It is easily seen in this case that

F " �² ~ �³ ~

~��

� �2

� �2

c ��

)c �

�

-

��

��

(symmetric)

(antisymmetric)(2.15)

where is the number of ways of representing as a sum of (distinct) positive� ²� ³ �� �

integers. The asymptotics of the combinatorial functions and are tabulated in books� �� �

of mathematical functions (see [AS]).

§2.3 Formalism of Asymptotic Spectral Densities

In many situations in which a self-adjoint operator has large or asymptotically infinitespectral density, use of a spectral measure on the real line is necessary for interpretationof sums indexed by the spectrum, or mediating more detailed spectral information. Aclassical example occurs in Sturm-Liouville systems on , whose “natural” spectral´�ÁB³measures for eigenfunction expansions are defined in terms of limits of spectral densities

of their restrictions to intervals (see [GS]). In the study of sums such as , a� ~ �� �

similar situation occurs if the spectrum of is very dense or continuous, e.g., if ¸, ¹ ( (�

is an elliptic differential operator on a “large” Riemannian manifold. See Simon'sreview article [Si, Sections C6, C7] for a perspective on this problem in Schrödinger

17

theory. Situations such as these certainly predominate in macroscopic systems. In suchcases, the appropriate probabilistic context for study of probability distributions incanonical ensembles involves integrals (rather than sums) of independent randomvariables with respect to appropriate spectral measures.

We begin with preliminary notions and facts. Henceforth will denote the greatest´ h µinteger function.

Definition 2.2: A is a -finite measure onspectral measure� � lb ~ ´�ÁBµÂ- ²,³ ~ ´�Á ,µ (� � is the of . The operator with point spectrumspectral function �

consisting of the discontinuities of , each discontinuity having multiplicity@ A-²,³�

,

@ A @ A-²,³ -², ³� �

cc

, is the - corresponding to .� �discrete operator

The operator has eigenvalue density essentially given by ( À��

�

The correspondence between spectral measures and nets is bijective.� ¸( ¹� ���

Indeed, the -net of functions converges uniformly to , determining .� � �@ A-²,³�

-

We now investigate when defines a canonical ensemble, using criteria established(�

in Chapter 1. We require the following convention: the domain of integration in aStieltjes integral includes endpoints, and

�

��²%³��²%³

� �� �c

� �

¦�c

�²%³��²%³ � �²%³��²%³À ²�À�³lim� �

We begin with a technical lemma.

Lemma 2.3. Let and be defined on be monotone non-increasing�² h ³ �² h ³ ´�Á �µÁ �² h ³and non-negative, be monotone and be Stieltjes-integrable with respect to�² h ³ �² h ³�² h ³ ´�² h ³µÀ and

Then

f f� �� �

� �

�²%³��²%³ c �²%³�´�²%³µ � �²�³À ²�À��³

This holds if “a” is replaced by “ ”, “ ” by “ ” or if is replaced by� � � ´�Á �µc c

´�ÁB³.

Proof: Assume without loss that is non-decreasing, and that is continuous on�² h ³ �² h ³´�Á �µ � ~ B; some inessential complications occur if the latter fails. Assume (otherwise� � and can be appropriately extended), and let

� � % � % � % � ÃÁ ²�À��³� � �

where are the discontinuity points of . If is empty the assertion is¸% ¹ ´�² h ³µ ¸% ¹� �

trivial. Otherwise let

18

: ~ ´�Á % µÁ : ²% Á % µÁ : ~ ²% Á % µÁà À� � � � � � � �

Then

� � �� �� : �

B B

�~� �~�

��²%³��²%³ ~ �²%³��²%³Â �²%³�´�²%³µ ~ �²% ³Á ²�À� ³�

and

�²% ³ � �²%³��²%³ � �²% ³ ²� ~ �Á �Áà ³À ²�À��³� �b�:

��b�

Hence

f f� ���~� �~�

�:

�²% ³ c �²%³��²%³ � �²�³À ²�À��³�

The fact that

�:�

�²%³��²%³ � �²�³ ²�À��³

and the first inequality in (2.20) imply (2.18), the remaining assertions follow similarly.Ê

Corollary 2.4: If instead of monotone is of bounded variation, then�

g g� �� �

� �

�²%³��²%³ c �²%³�´�²%³µ � ��²�³À

Theorem 2.5: Let be a spectral measure and the corresponding -discrete� �(�

operator. Let be a non-increasing function. Then is trace class if and only if�² h ³ �²( ³�

��

B

�²,³� ²,³ � BÀ ²�À��³�

Proof: If is trace class, then�²( ³�

tr �²( ³ ~ �²,³� Á- ²,³

� � @ A�

B

c �while

� � : ;� �

B B

�²,³� ²,³ ~ �²,³� À ²�À��³- ²,³

� ��c

If is bounded on , the last expression is finite. Otherwise, the lemma implies-² h ³ ´�ÁB³

19

� � �� �

� �@ A @ A� �

B Bc c

c c

�²,³� � �²,³� b �²� ³ ~ ² �²( ³³ b �²� ³ � BÀ-²,³ - ²,³

tr �

Conversely if (2.23) holds, then

tr �²( ³ ~ �²,³� � �²,³� b �²� ³ � BÀʲ�À�³- ²,³ - ²,³

� � �@ A : ;� �

B Bc

c c� �

Corollary 2.6: The operator is trace class for some if and only if it is for�²( ³ � �� �all .� � �

Corollary 2.7: The operator is trace class if and only if�c (� �

��

Bc ,� � ²,³ � BÀ� �

Definition 2.8: A spectral measure is 0- if 0.� �free ²¸�¹³ ~

Note that is 0-free if and only if is 0-free for all This fact together with� �( À�

Corollary 2.7 and Proposition 1.4 completely characterizes those spectral measures whichdefine canonical ensembles.

20

Chapter 3

Integrals of Independent Random Variables

In the limit of continuous spectrum for a Hamiltonian , sums of r.v.'s indexed by the(spectrum, such as value functions of quantizations of observables commuting with , are(most profitably represented as integrals for both notational and conceptual reasons. Wenow develop a theory and properties of these to the extent that they are useful later.These results will be in a more natural probabilistic setting in Chapters 7 and 8, whereintegrals of r.v.-valued functions over a measure space will be studied.

§3.1 General Theory of One Parameter Families

For , let be a one-parameter family of independent r.v.'s and a spectral, � � ?²,³ �measure on . We integrate with respect to as follows. For each letl � �b ? � �<� �~ ¸, ¹� ��1� be positive numbers (not necessarily distinct) such that the cardinality5 ²�Á �³ q ¸´�Á �µ¹� � of satisfies<

� ­ �5 ²�Á �³ ´�Á �µ ²�À��³��¦�

for all ( may be ). To avoid pathologies, assume also the existence of� � � � � � Bnumbers such that for any interval whose measure exceeds 4Á � � � 0 4Á

� �5 ²0³ � � ²0³À ²�À��³�

Definition 3.1: -net.¸ ¹< �� ��� is a spectral

Given a function ( on positive , we define the probability distribution� � �³

� �l �

�b

?²,³ ²� ³ � ² ³ ?², ³� � � �w- lim¦�

�

�

with sums on the right defined only if order-independent. The limit is in the topology ofconvergence in law, with the integral defined only if independent of .< �

If is real-valued on the Riemann-Stieltjes integral of with respect to can be� Á �l �b

defined as

9b

� �l �

�²,³� � �², ³ ² , ³ ²�À�³� � "lim²7 ³¦�

� �

with a partition of into intervals , , and the7 , , � , Á ²7 ³ ~ ² , ³l " " � � "b� � � � �sup

limit is defined only when independent of and , as well as order of summation (i.e.,7 ,�

convergence is absolute). In evaluating the right side of (3.2) and all similar sums we

21

apply the convention: if has an atom at , a partition may formally divide � , � � 7 ,into degenerate intervals , each consisting of just the point , such that¸ , ¹ ," � � 1��1

�� " �² , ³ ~ ²¸,¹³Â the union of such an interval with an adjacent non-degenerate

interval is also allowable. is an if , except possibly7 ², ³ ~ ², ³even partition � �� �

when is a rightmost non-empty interval, which is allowed smaller measure.,�

Theorem 3.2: If is a real number (i.e., a point mass) for each , then ?²,³ , ?²,³� �

coincides exactly with .9 ?²,³��

Proof: Assume that exists, and let be a spectral -net. For9 � � 1?²,³� ~ ¸, ¹� < �� � � �

� � 7 ~ ¸ , ¹h " l let be a sequence of even partitions of following the above� �� � 0b

� �

conventions, such that . The indexing set may be finite or the whole� ­ h²7 ³ � 0 �� ��¦B

of . For let satisfy˜h "� � 0 Á , Á , � ,� �� ����i

?², ³ c � ?²,³ � ?², ³ b ², � , ³Á ²�À�³� �

�� ����i

� ��� ��˜ "

and define

� ~² , ³

���

��0

��

��

�

� "

( ~ ?², ³ ² , ³Á ) ~ ?², ³ ² , ³À� �� � �� ��

� 0 � 0��i� �

� �� �

� " � "˜

If has compact support then is finite, and by (3.1) and (3.3), the distance between the� 0�

number and the interval approaches 0 as 0. Since� ����1

� � � � �

�

?², ³ ´( c � Á) b � µ ¦�

���¦B­

´( b � Á) c � µ ?²,³� Á� � � ��¦B

9­ ��and the result follows here, since the -net was arbitrary. For the general case, we� <�

must show

� ­�O, O��

�

��

O?², ³O �Á�� ¦ B

uniformly in . Let be an even partition of such that� " l7 ~ ¸ , ¹�b

��

� �O?², ³O ² , ³ � * ²�À�³� "

for any set of By (3.1b), � , À� �"

22

� " � "5 ² , ³ � � ²4Á ² , ³³Á� � �max

so that

� � "�, � ,

� �,� ,

�� � �"

�"

O?², ³O � � ²4Á ² , ³³ O?²,³OÀmax sup

Since is even and by (3.4),7

� � " ­� �O, O�� , q´�ÁBµ£

� �,� , �¦B

�� ��

O?², ³O � � O?²,³O ²4Á ² , ³³ �Á ²�À³�

" � "

sup max

convergence being manifestly uniform in .�Conversely, assume exists. Let be a sequence of partitions?²,³� 7 ~ ¸ , ¹� "� ��

of with and let . There exists an even partitionl � ­ "b� �� ��

�¦B²7 ³ �Á , � ,

7 ~ ¸ , ¹i i�" such that

8 ~ O?²,³O ² , ³ � BÂ�� ,� ,

�isup

" �i

� "

hence if , then� �²7 ³ � ²7 ³�i

�� ,� ,

��sup" ��

i

O?²,³O ² , ³ � �8À ²�À³� "

Let be chosen such that for each there is an integer (depending only< � � � �~ ¸, ¹ , �� �

on ) and a such that� �

�³ ¸� ¢ , ~ , ¹ ´ h µ The cardinality of is where denotes the greatest�� "

�� � �² , ³

�

�> ?� �

integer function, and for each for some .�Á , ~ , ��� � ��

�³ 1 ~ ¸� ¢ ² , ³ � � ¹ O?², ³O ² , ³ � If then .� �

�

� " � � "� � � � � �� 1

� � �

�

� ���

�³ � B� ­

� ¦ � monotonically.

Then if is the complement of 1 1� ��

g g g J B C Kg� � �� �

�

� � " � � "� "

�

� " �

� "

� � �

� � � � � � � � �� �

��1 ��1

� � � � � �

��1

��

?², ³ c ?², ³ ² , ³ ~ ?², ³ c ² , ³² , ³

� O?², ³O ² , ³ b O?², ³O

� b O?², ³O�

�

² ,

�

��

� � � �

�

�

� � �

�

�

�

�

��³

�²�À�³

� ¸� b �8¹ �Á�

�

�

�

­�¦ �

23

the above holding for sufficiently small. Thus exists and equals� �9?²,³�?²,³� Á Ê� completing the proof.

As is clear from the standard central limit theorem, expectation of r.v.'s does not scaleunder independent sums in the same way as some other “linear” quantities, such asstandard deviation. For this reason expectation will in general diverge in the formation ofintegrals (in which may be non-linear), while all other linear parameters converge to� �² ³define a limiting distribution. To avoid this inessential complication, we assumehenceforth that integrals involve -mean random variables.�

Definition 3.3: If is an r.v. or probability distribution, denotes the convolution of@ @ i�

@ � with itself times.

Recall that a distribution function is if for every and , there- � Á � � Á � � �stable � � � �

exist constants and such that� � � �

-²� % b � ³i- ²� % b � ³ ~ -²�% b �³À ²�À�³� � � �

Note that stability implies infinite divisibility..

Proposition 3.4: If exists and is non-zero, then is stable,@ ~ ?²,³ ²� ³ @ � �

� ~ ²� ~ �Á �Á �Áà ³² ³

²� ³�

¦�lim�

� �

� �

exists, and � @ ~ @ À�i�

Proof: Fix and let be a sequence such that . Consider a spectral -net� ¸ ¹ �� � �� �b� ���

< < <� � �~ ¸, ¹ �Á� ��1�� such that consists of with each element repeated times. For

� ��

� ~ �Á �Á �Áà Á let

@ ~ ²� ³ ?², ³��c� � � Á�

��1

� � �� �

�

�

�

@ ~ ² ³ ?², ³Á�� � Á�

��1

� � ���

��

where indicates a product.���

Then

@ ~ @ À² ³

²� ³�� ��c�

i� �

�

� �

� �

Letting , we conclude that� ¦ B

24

� ~² ³

²� ³�

�¦B

�

�lim

� �

� �

exists, and that � @ ~ @ À Ê�i�

Corollary 3.5: If of Proposition 3.4 has finite second moment then it is normal.@

Proof ¢ The normals are the only stable distributions with finite second moment (see[GK]). Ê

The following theorem generalizes Liapounov's theorem (see, e.g. [Ch]) to infinitesums of random variables, and will be instrumental in subsequent limit theorems. Recallthat convergence in law of a net of random variables or distributions is denoted¸? ¹� ��0

by is the normal distribution with mean and variance ? ¬ 4 5²�Á ³ � À�� �� �

Theorem 3.6: For each , let be independent zero-mean r.v.'s on the� � � ¸? ¹�� ����2�

same probability space and (or ). Le2 B 2 ~ B !� ��­¦�

� ; � ;� � ��� � �� � � �

�~ O? O Á ~ ? Á6 7 6 7� � � �� � � �� � � �

��2 �~2� �~ Á ~ À� �

� �

Then if the sum is asymptotically normal:�

� ��

�

��

­¦�

��1��Á ?�

�? ¬ 5²�Á �³À ²�À ³

���

���

�¦�

Proof: Let be an integer such that � � 2 ²�³� �

�8 9�

�~�

�

��

�~�

�

��

¦�

�

�

�

�

­

�

�

���

�

and ²��³

25

� 6 7�

�~� b�

2

��

�~�

�

��

¦�

�

�

�

�

�

­

�

�

��À ²�À��³

The sum is asymptotically normal by and Liapounov's theorem:��~�

�

�

�

? ²�³�

���~�

�

�

�~�

�

��

¦�

�

�

?

¬ 5²�Á �³À

�

�

�6 7���

By the same holds for²��³

; � À ²�À��³

?

�

�

�

�6� 9

�~�

�

�

�~�

2

��

�

�

�

��

The remainder

< � ²�À��³

?

�

�

�

�8 9��~� b�

2

�

�~�

2

��

�

�

�

�

��

satisfies . Hence; ­²< ³ �� �

�

¦�

8 � ; b < ¬ 5²�Á �³À Ê ²�À��³� � ��¦�

Theorem 3.7: If and , then9 9� �

B B� � 6 7; � ; �²? ³� O?O � � B

��

B

?²,³²� ³ ~ 5²�Á #³Á���

where .# ~ ²? ³�9 �

B � ; �

Proof: Let for be a spectral -net, and . Let and< � � �� � � � �~ ¸, ¹ � � ? � ?², ³� ��1 � � �

��� be as above. Then

26

�

�

; � ;

� � ;

� ­; �

; �

�

�

� �

� �

�

�

�

� �� �� �

� �� �� �

¦�

�

�

~ ~ � h ~ � À ²�À��³

O? O O? O

²? ³

O?O �

O?O �

� �6 7 6 78 9 8 9� �

6 76 6 7 7� � �

� � �

��

Hence

� � 6 7?²,³²� ³ ~ ? ~ h # ~ # 5²�Á �³ ~ 5²�Á #³Â

?

� �

�

� �

� � � �� � � �

��

��

lim lim� �

�

�

�

¦� ¦��

��

�

���

��

the second equality follows by Theorem 3.2, and the third by Theorem 3.6. Ê

§3.2 Background for Singular Integrals

Now we prove a result central to the singular integral theory in Chapters 4 and 5. Asingular integral is one where is singular. Since such integrals ?²,³ ²� ³ ²? ²,³³� � ; �

fail to exist in general under the current definition, we make one which is more useful.

Definition 3.8: Let be a one parameter family of independent r.v.'s on , and ?²,³ l �b

a -finite measure on Let , and for each , let � l � � <b B� �~�À - ²,³ ~ ²´�Á ,µ³ � � ~ ¸, ¹� �

be the ordered set of discontinuities of , a jump discontinuity of size being listed> ?-²,³�

�

� times. We define

:b

� �l �

�?²,³ ²� ³ � ² ³ ?², ³À ²�À�³� � � �lim¦�

�~�

B

�

Theorem 3.9: Suppose and are functions and� � � �² ³ ² ³

�

�­ ; � � ; � �

�

¦�

� � �

��Á O?O ²� ³ � BÁ � � ²? ³ ²� ³ � # � BÀ

: :� �6 7Then

: ? ²� ³ ~ 5²�Á #³À� �

Proof: By the hypotheses,

� � ;

� � ;

­

² ³ O? O

² ³ ²? ³

2 � B

� 6 76 7��

��

�

���

¦�

�

�

���

.

Hence

27

� 8 76 7��

��

���

¦�

;

;

­

O? O

²? ³

�Á ²�À�³

�

�

���

and is asymptotically normal by Theorem 3.6; the variance of is� �� �

� �? ² ³ ?� �� �

� � ;� �

��² ³ ²? ³ Ê�� , completing the proof.

Corollary 3.10: If and�� c � � �

: :� �6 7; � ; �O?O ²� ³ � BÁ # ~ ²? ³²� ³ £ �Á� � � ��

then

:� ?²� ³ ~ 5²�Á #³À� �

§3.3 Distributions Under Symmetric Statistics

Given a spectral measure and a spectral -net , the operator with spectrum � � < <� � �(induces a canonical ensemble under the conditions outlined in Chapter 2. For a net ofoperators , (with a function on ) we now consider asymptotics of* ~ �²( ³ �� � lb

corresponding value functions ; applications will be deferred until later.��

Definition 3.11: Let denote the functions of on with two continuous. ,� lb

derivatives near , locally of bounded variation, such that is non-% ~ � O�²,³O²� b �³�, c�

increasing for sufficiently large. For , those non-trivial spectral measures which, � � .�

are 0-free and satisfy

� � � ²,³� � ²,³ � B ²�À��³��

B� c ,� �

for are denoted by . Those whose spectral functions have� ~ �Á �Á �Á � 9 � 9�Á �Á� ��

three continuous derivatives near 0 comprise : À�Á�

The monotonicity condition on implies the same forO�²,³O²� b �³�, c�

O�²,³O²� c �³ À�, c�

The first and second conditions on guarantee existence of a²� ~ �³ � 9� �Á�

symmetric canonical ensemble, while the extra one on is required to control:�Á�

asymptotic behavior of . Let denote the spectral measures for which has an� < �� � ��Á

asymptotic distribution. Here and in Chapters 4 and 5 we show that .: � <�Á �Á� �

28

Proposition 3.12: Let and be non-decreasing and non-increasing,�² h ³ �² h ³respectively, with possibly infinite, and . Then�²�³ �²�³ ~ �

� � �²%³��²%³ c �²%³�´�²%³µ � �²%³��²%³Á ²�À��³� � �� � �

� � %Z

where

% ~ ² ¸% ¢ �²%³ � �¹Á �³ÁZ min inf

� B � � � � may be , and “ ” and “ ” may be replaced by “ ” and “ ”, respectively.c c

Proof: The proof is the same as that of Lemma 2.5 up to (2.20). We have

� � �� � ��~� �~� �~�: : :� � �

�²%³��²%³ � �²%³�´�²%³µ � �²%³��²%³Á ²�À� ³

since the fact implies that�²�³ ~ �

�:

�

�

�²%³��²%³ � �²% ³

for all , including . Equation (3.19) immediately gives (3.18) when . The� � ~ � � ~ Bcase of finite follows similarly, as do the replacements of “ ” and “ ” by “ ” and� � � �c

“ ”. � Êc

Proposition 3.13: Let be of bounded variation and non-increasing for large ,O�²%³O %with non-decreasing. Then�

� �@ A� �

� �

�²%³� ~ �²%³��²%³ b 6²�³ ² ¦ �³Á�²%³ �

� ��

where may be infinite, and are allowed.� � ¦ � Á � ¦ �c c

Proof: Assume . Since has bounded variation, it suffices to assume is� � B � �monotone. If is non-increasing, the result follows from Lemma 2.5, the general case�following similarly. If we may assume that is non-increasing, since any� ~ B O�Ointerval in which this fails may be disposed of by the above. In this case Lemma 2.5again applies, completing the proof. Ê

29

Letting above, we obtain�²%³ ~ %

� ��~ b�

�

�

> ?

> ?

�

�

�

�

�²� ³ ~ �²%³�% b 6²�³À ²�À��³�

��

Theorem 3.14: Let , and and one or both of the following hold:� � . � : Á� �� �Á

²�³ - ²�³ ~ - ²�³ ~ � � :Z ZZ�Á� �

²��³ �²�³ ~ � � 9 Á� �Á�

where is the spectral function of . Let be the -discrete operator corresponding to- (� ��

� !, and . Then the value function of satisfies� ~ �²( ³ � � ²* ³� � � �

l : ;��

� c ¬ 5²�Á #³Á�

��¦�

(3.21)

where

� ~ �²,³²� c �³ � Á # ~ �²,³� ²� c �³ � À ²�À��³� �� �

B B, c� , , c�� � �� �

Proof: ²�³ � Let denote occupation numbers on the symmetric canonical ensemble over��

( � then

� ~ �², ³�� � �� � �

and , where denotes the geometric r.v. with parameter . Let� � ²� ³ ²� ³ ��� � �

�c , c , c ,� ���

¸?²,³¹ ²� ³��

��c , denote independent r.v.'s, and�

?²,³ � ?²,³ c ²?²,³³À ²�À��³V ;

Then

:

� �

:� �� �

B B� � �; � ; ; �²O�²,³?²,³O ³²� ³ � O� ²,³O ²?²,³ b ²?²,³³³ ²� ³ À²�À��³V

A calculation shows

; ;H I H²?²,³ b ²?²,³³³ ~6²� ³ ², ¦ B³

6², ³ ², ¦ �³�

c ,

c�

�

.

Hence the right side of (3.24) is bounded by

30

� , ²� ³ b � � O� ²,³O²� ³ Á ²�À�³� ��

c� c , �B

:

� �

:� ��

�

�� �

where is chosen such that is not an atom of , for some� � �� � ²�³ ²��³2 � �Á - ²,³ � 2, ², � ³ ²���³ - , �� � �, and is continuous for . The second term

is 0 by Theorem 3.2. For , and thus, � Á ~ �� ��-², ³

� ��

, � Á�

2�� 8 9�

��

so that the first term is bounded by

lim lim lim ln� � �

��

¦� ¦� ¦�� �

, �

c�

�

c�� � � 2 � ~ 6² ³ ~ �À ²�À�³�

2� � � �

� � �� � �

� ��

��

� �8 9� �6 7

Hence, the left side of (3.24) is 0, while

� � ²�²,³?²,³³ � ~ � ²,³ ²?²,³³�V

~ � ²,³� ²� c �³ � � BÀ

: :

:

� �6 7�

� �

B B� �

�

B� , , c�

; � L �

�� �

By Corollary 3.10 with and � ~ � ~ �� � Á

:

���

�

B

�²,³?²,³²� ³ ~ 5²�Á #³Á ²�À��³V �

with

# ~ � À� ²,³�

²� c �³��

B � ,

, �

�

��

Thus is asymptotically normal.��To calculate the asymptotic expectation of , we assume initially that .� �²�³ � ��

Then

;²� ³ ~ ~ b Á�², ³ �², ³

²� c �³ ²� c �³�

� �

� �� �

� � �: ;�

� �

, ,, � , �

� �

� �

� �

� �

with chosen so that it is not an atom of , and the absolute value of the summand is� �decreasing in for . By Proposition (3.13) the second sum on the right is, , �� �� � �

31

:� �B C� �

� �

B B

, ,

�²,³ - ²,³ � �²,³

� c � � c �� ~ � b 6²�³Á ²�À��³

� ��

while

� � �B C, �

�

,� �

, ,�

�

�

�

�

�

�

� �

� �

�², ³

²� c �³ ²� c �³ ²� c �³~ � ~ � b �² ³Á ²�À� ³

�²,³ �²,³ � �²,³

� �� �

where

O�² ³O � � Á� O�²,³O

²� c �³� �

���

, ² ³

,

Z �

�

and . If is 0 in a neighborhood of 0 then (3.28) holds, ² ³ � , ¢ � � -Z Z� inf F G�², ³Z

�

for . Otherwise, and� �~ � , ² ³ �Z­

� ¦ �

O�² ³O � �-²,³ � �-²,³ ~ � Á� � � � � �

²� c �³ , , ² ³� �

� � � �� � �� � �

, ² ³ , ² ³ -², ² ³³

, Z�

Z� � �� � �

�

for some and sufficiently small. The equality obtains as follows. Let be thrice� � � -�continuously differentiable for is an inverse of since , � À -² h ³ , ² h ³Á -k, ² ³ ~� � �� �

²� � � -² ³³ - ²�Á µ� � � (note that is not generally invertible on , since it may be constanton intervals). Hence equality follows formally by the change of variables ; a�Z ~ -²,³rigorous argument involves the definition of the Stieltjes integral as a limit of sums overpartitions. Since -²,³ � 2, ², � ²�Á µ³Á� �

, ² ³ � ²� � � -² ³³À2

� � � ��8 9

��

Hence,

��

-², ² ³³

�Z

Z�

�� �� �� �

� �

� � �2 �2

, ² ³ � �� � -², ² ³³ ~ À ²�À��³

�� � �

Combining (3.28-30),

; � � � �� �

²� ³ ~ � b � ~ b6 ² ¦ �³À ²�À��³� �²,³ �

� c �� �� 6 7 6 7

�

B

,c c� �

� �

This calculation is much simpler if (and hence near zero). By�²�³ ~ � �²,³ ~ 6²,³(3.27) and (3.31),

l l l8 9 8 9� � ; � ;� �

� c ~ ²� c ²� ³³ b ²� ³ c� �

� � � �

32

¬ �²,³?²,³²� ³ b � ~ 5²�Á #³ÁV�¦� �

B

:

��� �

completing the proof of .²�³

In this case , and²��³ �²,³ ~ 6²,³ ², ¦ �³

;²O�²,³?²,³O ³ ~ ²�À��³V 6²� ³Â ², ¦ B³6²�³Â ², ¦ �³

�c ,H �

with the same bounds for . Hence asymptotic normality of follows by;²� ²,³? ²,³³ �V��

�

Corollary 3.10, with . By Proposition 3.13,� ~ �Á � ~ ��

; �� �

��

²� ³ ~ � ~ � b 6²�³Â�²,³ - ²,³ � �²,³

� c � � c �

= ²� ³ ~ � b 6²�³Á� �²,³�

²� c �³

� � �

�

�

�

� �B C�� �

B B

, ,

�

B ,

, �

completing the proof. Ê

§3.4 Antisymmetric Distributions

Definition 3.15: The class is defined in the same way as (Definition 3.11),/ .� �

without the condition on differentiability. For is without the 0-� � / Á ; 9� � ��Á �Á

freedom condition.

The relaxed conditions above still allow proof of central limit theorems for observablesunder antisymmetric statistics, since antisymmetric occupation numbers form a non-singular family. In this case is Bernoulli (see §2.2), with moments easily bounded by���

; ;²O� O ³ � � b ²� b �³ ~ 6²� ³ ²� ~ �Á �Á �Áà ³ ²�À��³V� �� �

� �� , c� c ,

�

: ;8 9� �� �

where

� � � c ²� ³ ~ � c � b � ÀV� � � ��

� � � �,

c�

; 6 7��

By Corollary 3.10, is asymptotically normal. By Proposition 3.13,� ~ �², ³�� � ��

� �

33

; �� �

�� �

²� ³ ~ � ~ � b 6²�³ ²�À��³�²,³ - ²,³ � �²,³

� b � � b �

= ²� ³ ~ � b 6²�³ ² ¦ �³À� �²,³�

²� b �³

� � �

�

�

�

� �: ; B C : ;�� �

B B

, ,

�

B ,

, �

Applying Proposition 3.13 to (3.34), we have

Theorem 3.16: If and then� � / � ; Á� �� �Á

l : ;��

� c ¬ 5²�Á #³Á�

�

where

� ~ � Á # ~ ��²,³ �²,³�

²� b �³ ²� b �³� �� �

B B

, , �

,

� �

�

� �.

34

Chapter 4

Singular Central Limit Theorems, I

The integration theory of Chapter 3 deals with non-singular integrals in that, e.g.,

: ; �²? ³�� is finite. This section, from a probabilistic standpoint, is an illustration of

the singular theory for the family of geometric r.v.'s arising under symmetric statistics.The non-normality of these integrals, demonstrated in Chapter 5, is closely connected tothe "infrared catastrophe" of quantum electrodynamics.

The most convenient formulation (with a view toward applications) is in the languageof limits rather than integrals. By Theorem 3.14, we may assume that .�²�³ � �

§4.1 The Case -²,³ ~ �, Á �²,³ � ��

With the assumptions of §3.3 still in force, this falls into the category of “weakest”violations of the hypotheses of Theorem 3.14. We have

, ~  = ²� ³ ~ �² ² ³³ b 6 Á ²�À�³� �

�� ��

c�n 8 9��� �

�

where

� ��

�² ³ ~

O On ln

(a definition which will hold henceforth); hence Theorem 3.14 cannot hold here. Theasymptotic distribution of , though normal, has a new nature in that, after normalization��to standard variance and mean, is asymptotically dominated by occupation numbers��� ,� �� � whose spectral values are arbitrarily small. Normality is in fact not universal inthis situation, as will be seen later.

Note that in all “singular” sums of r.v.'s to be considered, leading asymptotics will��depend only on spectral behavior near the origin; parameters related to global spectraldensity will not contribute. We now consider a prototype.

Lemma 4.1: , , If and then-²,³ ~ �, ², � �³ �²,³ ~ ��

9 � ² ³ � c ¬ 5 �Á À ²�À�³�

�� �� �

�

�� �8 9 8 9�

� �

Proof: Assume without loss that . The characteristic function of is� ~ � 9�

35

) ; ;9�9 ! �! ² ³�

c�!

c�!

�~�

B c �

²c � b� ² ³!³

�� �

� � �

��

� � �

��

²!³ ~ ²� ³ ~ � �

~ � Á� c �

� c �

8 9

8 9 ll

² ³ �

� �

² ³ �

� �

8 9�J K

� �

� �

� � � �

so that

ln ln ln)� � �

��9

�

��~�

Bc � ²c �b� ² ³!³

�²!³ ~ b � c � c � c � À

c �! ² ³

��J K6 7 6 7� � � � � �l l

²�À�³The branch of the logarithm is determined by analytic continuation from the real axis for� large. We rewrite (4.3) as

ln)� � �

9

�

�²!³ ~

c �! ² ³����

b � c � c ² �³�6 6 7 7l�~�

c �

> ?l

��

ln ln� � � �

c � c � c ² � c � ² ³!³�6 6 7 7l�~�

²c �b� ² ³!³

> ?l

��

ln ln� � � � � � � �

b � c ² � c � ² ³!³�J K6 7l l�~�

> ?��

ln ln� � � � � �

b � c � c � c �� J K6 7 6 7�~ b�

Bc � ²c �b� ² ³!³

> ?l l

��

ln ln� � � � � �

~ �² �³ c �² � c � ² ³!³�J Kl l�~�

> ?��

� � � � � �

b ² �³ c ² � c � ² ³!³ ²�À�³�J Kl l�~�

> ?��

ln ln� � � � � �

36

b �² �³ c �² � c � ² ³!³ Á� J Kl l�~ b�

B

> ?��

� � � � � �

where

�²%³ ~  �²%³ ~ ²� c � ³À� c �

%ln ln8 9c%

c%

In the first sum ranges from to , so that�� � �

�J Kl l

J K� l�~�

�~�

Z

> ?

> ?

�

�

�

�

�² �³ c �² � c � ² ³!³ ²�À³

~ � ² �³²� ² ³!³ ²� b 6² ² ³³³ ² ¦ �³À

� � � � � �

� � � � � � �

Similarly

� J Kl l

J K� l�~ b�

B

�~ b�

BZ

> ?

> ?

�

�

�

�

�² �³ c ² � c � ² ³!³ ²�À³

~ � ² �³� ² ³! ²� b 6² ² ³³³ ² ¨ �³À

� � � � � �

� � � � � � �

ln

Note that is bounded and monotone decreasing for ,� ² %³ ~ % � ´�Á �µZ %b�c�

%²� c�³�l �

�

ll

�

�

l

l

%

%

as is , for . Thus (3.20) applies to the sums in (4.5) and� ² %³ ~ % � ´�ÁB³Z �

� c��l �l%

(4.6), and (4.4) becomes

ln) � � � � �� � �

�� �9

�

��

�Z

�²!³ ~ b � ² % ³�% b 6²�³ ²� b 6² ² ³³³� ² ³!

c �! ² ³ �

� J K� l

b � ² %³�% b 6²�³ ²� b 6² ² ³³³� ² ³!�J K� l�

� � � � ��

BZ

b ² �³ c ² � c � ² ³!³ ²�À�³�J Kl l�~�

> ?��

ln ln� � � � � �

~ b �% c �% ²� b 6² ² ³³³c �! ² ³ � ² ³! � �

� � c � %

� � � � �

�� � �� �

�

�� �

B �

%J K� � l�l

37

c � c ² ¦ �³À� ² ³!

�� 8 9l�~�

> ?��

ln� �

� ��

For small the arguments of all logarithms are near , so that branches are defined� l ]b �by analytic continuation.

For and , where is bounded.% � O%O � Á ²� c %³ ~ c % c ²� b %:²%³³ :² h ³] � %� �ln

�

Noting that

� � � �

� � � �­

² ³! ² ³!

�� � ²�À�³l l �¦�

we have

� � �: ; : ; : ;l l�~� �~� �~�

� �

�

> ? > ? > ?� � �� � �

ln � c ~ c b ²� b �²�³³� ² ³! � ² ³! ² ³ !

� � � �

� � � � � �

� � � � � �

~ b ²� b �²�³³ ²�À ³c � ² ³! � ² ³ ! �

� � �

� � � �

� � ��l l� �: ; : ;�~� �~�

� �

�

> ? > ?� �� �

~ � b 6²�³ b b 6²�³ ²� b �²�³³ ² ¦ �³Àc � ² ³! � ² ³ ! �

�

� � � �

� � � �� ��l : B C ; : B C ;

qqqp� �

�ln

Finally, replacing by we have> ?� �� �

� : ;l�~�

�

�

> ?��

ln � c ~ b b �²�³ ² ¦ �³Á² ³! c �� ² ³! !

� �

� � � �

� � �� ��

thus

ln)� � � � � � � �

�� � � � �� �9

� � �

� � ��²!³ ~ c b c b c b �²�³ ²�À��³

�! ² ³ � ² ³! � �� ² ³! !

� � �J K

~ c b �²�³ ² ¦ �³À!

�

�

���

Since is the characteristic function of , the Lévy-Cramér convergence� 5 �Ác !�

�� 6 7���

theorem completes the proof. Ê

38

§4.2 The Case ²�- ³ ²�³ � �Á �- ²�³ ~ �Z Z Z

In this case the “leading behavior” of arises from its small terms and can be� ,� ��

analyzed with use of the previous lemma. Recall that the convolution of distributionfunctions is defined by

²- i- ³²%³ ~ - ²% c &³�- ²&³Á ²�À��³� � � ��with the obvious extension to probability laws.

Lemma 4.2: Let be a collection of independent r.v.'s and be another.¸? ¹ ¸? ¹� �~� �~�B Z B

�

Let and converge a.e. If for all then: ~ ? : ~ ? - � - ²%³ %Á� �� ?Z Z

� ?� �Z

- ²%³ � - ²%³À: :Z

Proof: This follows by direct calculation for finite sums, and in the general case from thefact that almost sure implies distributional convergence. Ê

The sets and are defined in Definition 3.11.. :� ��Á

Theorem 4.3: Let and , with and the spectral function of .� � . � : �²�³ � � -� �� ��Á

Assume that and . Then- ²�³ ~ � - ²�³ ~ �� � �Z ZZ

� �� �

² ³ � c ¬ 5 �Á Á� ��²�³8 9 8 9� �

with given by (3.22).�

Proof: We prove this in three parts, first assuming , and then� � �Á - ²� ³ � �ZZZ b

sequentially eliminating the second and first conditions.

I. Assume

- ²� ³ � �Á �²,³ � �ÀZZZ b

Then

� ~ � ~ � Á� � ���~�

B

�

and since� � : Á�Á�

- ²,³ � � ²� � , � ³ ²�À��³ZZZ �

for some . Let� � �

39

- ²,³ ~ Á ²�À��³- ²,³Â , �-² ³Â , �

� H �

� �

�� � � �� � be the spectral measure defined by and the corresponding total- ² h ³Á � ~ ��

��

occupation number. Define

/ ²,³ ~ b Á ²�À��³- ²,³ �, c - ²,³

�

� �B C B C� �

�

and let be the ordered discontinuities of , the size of a<� � �i i

� ����B~ ¸, ¹ / ² h ³

discontinuity being its multiplicity in . Let be the discontinuities of< <� � �i � �

� ����B~ ¸, ¹

> ?�,�

�, so that

, ~ À ²�À�³�

���� n �

Let and be self-adjoint operators in Hilbert space with spectra and ,( (� � � �i � i �< <

respectively, again with multiplicity. Note that

� � c/ ²,³ � � ², � ³Á ²�À�³�,B C�� l�

since for ,�Á � � l

� � ´� b �µ c ¸´�µ b ´�µ¹ � �À

Furthermore, if , then, for ,, � , , � , � ,� �� �²�c�³ ²�c�³i � � i

� �

@ A�,� �Á / ²,³ � � c �Á

�

��

contradicting (4.16). Thus,

, � , ²� � � � B³À� �²�c�³ �i � (4.17)

Also, by (4.16),

, � , ²� � � � B³À ²�À��³� �� �� i

Let and be the symmetric occupation number r.v.'s¸� ¹ ¸� ¹� ��i �

����B ����B�

corresponding to and and( ( Á� �i �

� ~ � Á � ~ � ²�À� ³� � � �i i � �

�~� �~�

B B

� �� �total number r.v.'s. By Lemma 4.1 and the definition of ( Á�

�

40

8 � � c ¬ 5²�Á �³À ²�À��³² ³ �

� �� ��

� ��

� ¦�

�� � �

��l : ;Note also that by (4.15),

; L ;�� � � � �

� � � �

� �

�­

J: ; Kl H I6 7: ;l6 7 6 7l l

8 9

² ³ ² ³

�� ~ ²� ³ b ²� ³

�

~ b² ³ �

�

² ³

² ³ c � ² ³ c �

~ 6 �Á�

O O

� � �

�

� �

� �� � �

�� �

�

� ��

� �

� �

exp

exp exp

ln �¦ �

so that

> � � ¬ Á ²�À��³² ³

�� ��

� ��

¦��

�� ��l

with the point mass at 0. If we define��

� ~ � c � Á� � ��Á� �

�

then (4.21) and (4.20) imply

8 � 8 c> ~ � c ¬ 5²�Á �³À ²�À��³² ³ �

. �� � � ��

�Á� � � �Á��

� ¦�

�� � �

��l : ;The d.f. of is���

�

- ²%³ ~ ²� ~ �³ ~ � c � � ~ � c � ²% � ³Á� �

�~� �~�

´%µ ´%µc , c �, c ´%b�µ,

�

� � �

�� � � �

� � �� �8 9F l�� � �

which is monotonic in E . By (4.17) and (4.18),���

- ²%³ � - ²%³ ²� � �³ ²�À��³

- ²%³ � - ²%³ ²� � �³ À

� �

� �

� �

� �

²�c�³i

��

� �� i

We decompose

< < <� � �i i0 i00~ r Á

where the set corresponds to the discontinuities in and to those in< <� ��i0 i00- ²,³> ?�

> ?�, c- ²,³ i0 i00 i� �

� � � �; the intersection of and is listed twice in . Let< < <

41

: ~ ¸� ¢ , � ¹Á : ~ ¸� ¢ , � ¹Á ²�À��³� � � � � �0 i i0 00 i i00

� �< <

and define

� ~ � Á � ~ � À ²�À�³� � � �i0 i i00 i

��: ��:

� �� �� �0 00

Note that is the total occupation number corresponding to the d.f.� �� �i0 i006 7

- ²,³ ~ - ²,³ ~ ²�, c - ²,³³ À- ²,³ �

� �

��i0 i00 �

� �8 9

By their definitions, and , are identically distributed.� �� �� i0

By (4.12-13), is monotone non-decreasing. Since ,- ² h ³ � :i00�Á� �

- � : Ái00��

and

²- ³ ²� ³ ~ ²- ³ ²� ³ ~ �À ²�À�³i00 Z b i00 ZZ b

By Theorem 3.14 and the aboveÁ

8 � � c ¬ 5²�Á �³Á#

�� �i00 i00

i00

i00n : ;�

�

where

� ~ �²�, c - ²,³³Á�

� c �

# ~ �²�, c - ²,³³À�

²� c �³

i00 �

�

B

,

i00 �

�

B ,

, �

��

�

�

�

�

�

The r.v.

� ~ � ²�À��³� ��Á� �

�~�

B

��has term while the term of in (4.19) is . Thus by (4.23) the d.f. of the� � Á � � �!� � !� i i

²�c�³ �� �

� � �!� �Á� i� term of is bounded from above by that of the corresponding term in . Hence by� �

Lemma 4.2, letting denote the d.f. of ,- ²%³ ��

- ²%³ � - ²%³Á� �� �i �Á�

and similarly, by (4.24),

42

- ²%³ � - ²%³À� �� �� i

Since these inequalities are preserved under renormalization of expectations andvariances, we have by (4.20)

- ²%³ � - ²%³Â - ²%³ � - ²%³Á ²�À��³8 8 8 8� � � �

� i i �Á�

where

8 ~ � c À²,³ �

� �� �i i

�

�

�� �

��l : ;Thus by (4.28), and (4.22),

- ²%³ � - ²%³ � - ²%³ - ²%³Á ²�À� ³8 8 8 ¦�5²�Á�³� � �

� i �Á� ­�

and by (4.29),

8 ¬ 5²�Á �³À ²�À��³�i

But by the definition of ,8�i

8 ~ � c b � c ²�À��³² ³ � c � � ² ³ �

� ��

~ � c b 8 À² ³ � c � � # ² ³

� ��

� �

� �

i i0 i00� � i00 i00

�

i0 i00� � i00

�

i00

�� � � � �� �

�� �

�� � � � � � �

�� �

l l: ; : ;l l: ; l

The coefficient of approaches 0 and so the second term converges in law to . Thus8�i00

��

by (4.31),(4.30), the independence of and , and the identity of and ,� � � �� � � ��i0 i00 i0

�� � � �

��

² ³ � c � �

�� c ¬ 5²�Á �³À

�l : ;��

�

� � i00

� ¦�

Now let

� ~ � c � À ²�À��³� ����

Using by now standard arguments, we conclude

n : ;� �

� ��

# ² ³� ² ³ c ¬ 5²�Á �³Á ²�À��³

� ² ³�

��

�

where

� ² ³ ~ �-²,³Â # ² ³ ~ �-²,³À� �

� c � ²� c �³� �

B B

, , �

,

� �� �� �

� �

�

Equation (4.32) implies

43

�� � �� � � �

� ���

�� � ��

� � ��

² ³ � ² ³ � c � �

� �� c ~ � ² ³ c ²�À��³

�

b � ² ³ c b c À² ³ � � �

� � �

l l: ; : ;l : : ;;

� ��

�

� � i00

�

�i00 �

By the definitions,

� b�c ~ �-²,³ ~ � ² ³Á ²�À�³� �

� � c �i00 �

�

� ,

B�

���

��

since

��

B

, ��

�� �

� c � ��²�, ³ ~ À

�

�

�

Because , the last term in (4.34) converges to by (4.33) and (4.35). On the� �

�

² ³�l ¦ � �

other hand, the first term on the right of (4.34) converges to the standard normal, so that

�� �

�

² ³ �

�� c ¬ 5²�Á �³ ²�À�³l : ;�

�¦�

completing part I.

II. We now eliminate the restriction on and assume only that . In this- ²�³ �²,³ � �ZZZ

case there clearly exists a spectral function corresponding to a distribution- ² h ³�

� l� ��Á� : ²�³ - ²,³ c - ²,³ , � ²�³� such that is monotone non-decreasing for ,

²- ³ ²� ³ ~ �Á ²�³ ²- ³ ²� ³ ~ � ²�³ ²- ³ ²� ³ � �� Z b � ZZ b � ZZZ b, and . The construction of such aspectral function involves finding one with sufficiently negative third derivative which isconstant for all values of larger than some sufficiently small number . The, , � ��

function satisfies the hypotheses of Part I.- �

Analogously to (4.14), define

- ~ b À ²�À��³- ²,³ - ²,³ c - ²,³

�i

� �B C B C� �

Let be the ordered set of discontinuities of and < < <� � � � �i i i

� ����B � ����B~ ¸, ¹ ~ ¸, ¹

that of . As before> ?-²,³�

� � c - ²,³ � � ², � ³Á- ²,³B C�

l�i

so that

44

, � ,  , � , À ²�À��³� �� �� �� ²�c�³i i

Let

� ~ � Á � ~ � ²�À� ³� � � �� ��~� �~�

B B

�i i

�

be the total symmetric number of r.v.'s obtained from and , respectively, and-²,³ - ²,³i

� ~ � c �� � ��

�. As before we conclude from (4.38) that

- ²%³ � - ²%³Â - ²%³ � - ²%³Á ²�À��³9 9 9 9� � � �i i �

where

9 ~ � c Á² ³ �

�� �

� � �

�l : ;with and corresponding similarly to and , respectively.9 9 � �� � � �

i � i �

We make the decomposition , where corresponds to< < < <� � � �i i0 i00 i0~ q

discontinuities in and to those in . Let and be defined> ? > ?- ²,³ -²,³c- ²,³i00 i0 i00� �

� �� � �< � �

analogously to (4.25). By Theorem 3.14,

n : ;�

�#� c ¬ 5²�Á �³Á ²�À��³

�i00

i00i00

�

with

# ~ �²- ²,³ c - ²,³³Â ²�À��³�

²� c �³

� ~ �²- ²,³ c - ²,³³À�

� c �

i00 �

�

B ,

, �

i00 �

�

B

,

��

�

�

�

By the conclusion of Part I,

�� �

�

² ³ �

�� c ¬ 5²�Á �³Ál : ;�i0

i0

where

� ~ �- ²,³À�

� c �i0 �

�

B

,� �

Hence

9 ¬ 5²�Á �³À ²�À��³�i

As in (4.21),

45

> � � ¬ À ²�À��³² ³

�� �

� � ��l � �

Since is continuous, a net of probability distributions which converges to- ² h ³5²�Á�³

- ² h ³ %5²�Á�³ pointwise does so uniformly in . Thus by (4.44),

� O- ²%³ c - ²%³ O �Á�l �

sup%�

5²�Á�³ 9¦��

i ­

and

O- i- ²%³ c - i- ²%³ ~ ²- ²% c &³ c - ²% c &³³�- ²&³9 > > 9 >5²�Á�³ 5²�Á�³cB

B

� �� � �i i| �

²�À�³

� �- ²&³ ~ �À � ��cB

B

>�­�¦ �

Also by (4.44),

- i- ²%³ - ²%³Á5²�Á�³ 5²�Á�³>�­�¦ �

so that

- i- ²%³ - ²%³À ²�À�³9 >¦�

5²�Á�³� �i ­

�

Thus Lemma 4.2, (4.44), (4.40), and (4.43) imply

- i- ²%³ � - i- ²%³ ~ - ²%³ � - ²%³ - ²%³ ²% � ³Á9 > > 9 99 5²�Á�³¦�� �� � ��

i i� ­ l�

so that, with (4.46), we have

9 ¬ 5²�Á �³À ²�À��³�

III. In the general case we may assume , by Theorem 3.14. Then�²�³ £ �

� ~ �²�³� b ²� c �²�³� ³À� � � �

By Theorem 3.14 (with a slight modification if the integrand of below is not��

asymptotically monotone),

n : ;6 7�

�#� c �²�³ � c ¬ 5²�Á �³Á

�

�

�

¦�� �

�

where

� ~ �  # ~ � À²�²,³ c �²�³³ ²�²,³ c �²�³³ �

� c � ²� c �³� �

� �

B B

, , �

� ,� �� �

�

� �

46

Together with the result of Part II, this gives

l : ; : ;�� �

� c ¬ 5 �Á Á� �²�³�

� �

completing the proof. Ê

47

Chapter 5

Singular Central Limit Theorems, II

This chapter deals with the most singular families of geometric r.v.'s arising under ourhypotheses, those from spectral measures with non-vanishing density near 0. Thesingularity of this situation fundamentally changes both the results and the proof of theassociated limit theorem. The normal distributions of integrals appearing in previouscases are now replaced by an extreme value distribution whose parameters are entirelydetermined by spectral density at 0. This will be shown in the prototypical case� ²,³ ~ �-²,³ ~ �,Á� and in general through an approach like that of Chapter 4. Theapproximately linear behavior of at 0 will be exploited to decompose into two-²,³ ��independent r.v.'s, to the first of which the prototypical case will apply, and to the secondTheorem 4.3.

The is that with d.f. . The normalized distribution,extreme value distribution �c�c%

with zero mean and unit variance, has d.f.

exp8 9c � Á ²À�³c %b6 7�

l�

where is Euler's constant. For applications of this distribution to the� ~ À��Ãstatistics of extremes, see [G].

§5.1 Asymptotics Under Uniform Spectral Density

In this section and are defined as before. We will require the following simple� Á Á �� �lemma.

Lemma 5.1: Let be a net of probability d.f.'s, and¸- ² h ³¹� ���

- ²%³ - ²%³ ²% � ³Á ²À�³��­ l¦�

where is a continuous d.f. If is any net of d.f.'s, then-² h ³ ¸. ² h ³¹� ���

lim sup lim sup� �

� � �¦� ¦�

- ²%³ i. ²%³ ~ -²%³ i. ²%³ ²% � ³Â ²À�³ l

(5.3) holds as well for the lim inf.

Lemma 5.2: If for some , and then-²,³ ~ �, � � � �²,³ � �Á

48

9 � � c ¬ � À�

� �

��

��

�: ;e 6 7eln �

¦�

c�c%��

Proof: We may assume . The logarithm of the ch.f. of is� ~ � 9�

ln ln lnln

) ;��

�9

�! 9 b

�~�

B c �

c �b� !�

���

�²!³ ~ � ~ �! b Á ²À�³² ³ � c �

� c �: : ;; : ;�6 7���

�� �

ln² ³

where the branch of each term is principal for , and determined by analytic! � �continuation from elsewhere. Fixing ! � � !Á

lnln

)��

�9�²!³ ~

�! ² ³

b ² ²� c � ³ c ² �³ c ²� c � ³ b ² � c �! ³³

b ² ² �³ c ² � c �! ³³

b ²� c � ³ c � c �

�

�� 8 8

�~�

c � c �b�!

�~�

�~ b�

c � ²c �b�

> ?

> ?

> ?

�

�

�

��

��

��

ln ln ln ln

ln ln

ln ln

�� �� �

�� �� �

�� �� �

�� �� �

!³99À

²À³

Let

�²%³ ~ Á �²%³ ~ ²� c � ³Á� c �

%ln ln: ;c%

c%

with branches as before. The function is analytic at all points within unit distance�² h ³of the real axis interval and is analytic at all points within distance of´�Á �µÁ �² h ³ �

�

´�ÁB³ + + ! £ �. Denote these regions of analyticity by and , respectively. Let and� �

O !O � % � ´�ÁB³� �� . If , then

�²% c � !³ ~ �²%³ c � !� ²%³ b 9²%³Á� � Z

where

g g g g g g6 76 7 6 7

6 7 6 78 99²%³ � �

²� !³ � ²%³ � ! � !~ c ~ c

b6

� � �� Z

�c��c�

² !³ ² !³� c� � c�

�c� �²� c�³ � c�

�

lnc²%c� !³

c%

� �

% %

� ! c� !c�

% %�

� �

� �

²À³

49

� b6²�³À

688 996 7� c�� c�

�

² !³� c�

c� !

%

�

%

�

�

The term satisfies66 7� c�� c�

c� !

%

�

6� *Á

6 76 7

� c�� c�

� c�� c�

c� !

%

c� !

%

�

�

for some and sufficiently small values of the argument, and* � �

6²�³ ~ c� c � �

² !³ !

c� !

�

�

� �(5.7)

remains bounded for all , uniformly in (in fact independently of ). Thus if� � � % � + %�

! £ � % % � + is fixed, is bounded uniformly in for and sufficiently small, and9²%³²� !³ � ²%³ �� � Z �

�

J K J K� ��~ b�

Bc � c �b� !

�~ b� �~ b�

B BZ

�

> ?

> ? > ?

�

� �

��

�� ��

² ²� c � ³ c ²� c � ³³ ²À�³

~ � !� ² �³ ²� b 6² ³³ ~ ²� b 6² ³³ ² ¦ �³Á� !

� c �

ln ln�� �� �

��� �� � � �

�

which clearly also holds if .! ~ �Similarly, one can show that for ,! � l

�

J : ;K��~�

c � c �b� !

�~��

> ?

> ?

�

�

��

��

² ²� c � ³ c ² �³ c ²� c � ³ b ² � c � !³³ ²À ³

~ � ! c ²� b 6² ³³ ² ¦ �³À� �

� c � �

ln ln ln ln�� �� �

��

�� �� �

� � ���

The summands in the last terms of (5.8) and (5.9) are decreasing in , and Proposition���3.13 implies

��~ b�

B

�c�

> ?���

� ! �!

� c �~ c ²� c � ³ b �²�³ ²À��³

�

���ln

and

50

� ! c ~ ²� c � ³ b �²�³ ² ¦ �³À ²À��³� � �!

� c � �� �

�� ��: ;�~�

�c�

> ?���

��ln

Finally,

� � �: ; : : ;;:B C ; :B C ; : ;

�~� �~� �~�

> ? > ? > ?� � ��� �� ��

² ² �³³ c ² � c � !³ ~ � c � c ²À��³�!

~ b � c c b � b � c Á� � �! �!

ln ln ln ln

ln ln ln

�� �� ��

! ! !�� �� � �

where we continue our convention on branches. Stirling's expansion gives

ln ln ln! �²'³ ~ ' c ' c ' b ²� ³ b 6 Á ²À��³� � �

� � O'O: ; : ;uniformly for arg . Thus' � �" �

ln ln ln! ! !� �� �� �: ; :B C ; :B C ;� c b b � c c b ��! � � �!

~ � c b b b � c c b ��! � � � � �!

�ln ln ln!

� �� �� �� �: ; :B C ;J :B C ; :B C ;K

b c b � c b6² ³ ²À��³�! � �! �!

~ � c c b c b � b 6² ³�! � � �! �

�

� �� � ��

! �� �� � ��

ln

ln

:B C ;: ; :B C ;: :B C ; ;

c�

�

c ² ³ c b 6² ³�! �!

� ��� �ln

~ � c c ² ³ b 6² ³ ² ¦ �³À�! �!

ln ln! �� � �� �: ;

Combining (5.5) and (5.8-14)

ln ln² ²!³³ ~ � c b �²�³Á�!

) !�

9� : ;

51

so that

) ! ­ !� �

9¦��

²!³ ~ � c ²� b �²�³³ � c À ²À�³�! �!: ; : ;�

The right side is the Fourier transform of the distribution with d.f. , and the proof is�c�c %�

completed by the Lévy-Cramér convergence theorem. Ê

§5.2 Asymptotics for Non-Vanishing Densities Near Zero

Theorem 5.3: If , and , then� � : - ²�³ ~ � � �Á �²�³ � ��ÁZ

�

9 � � c ¬ ��² ³

� ��

��

�: ; ¦�

c�c

%�²�³��

where

�² ³ ~ �b ²À�³�²�³�

�

� ~ �²- ²,³ c �,³ b �-²,³À�²�³ �²,³ c �²�³

� c � � c �

��

��g : ;g� �

ln ,

� �

B B

, ,� �

Proof: I. We assume initially that and�²,³ � �

- ²,³ � � ²� � , � ³ ²À��³ZZ �

for a . Let be as in 4.13, and� � � - ²�³ ~ c �Á - ²,³ZZ �

/ ~ b À ²À��³- ²,³ �, c - ²,³

�

� �B C B C� �

Let be as in Theorem 4.3, and be the discontinuities< <� � � �i i � �

� ����B ����B�~ ¸, ¹ ~ ¸, ¹

of i.e., .> ?�,�� �

�� ��Á , ~

We have

� � c/ ²,³ � � ², � ³ ²À� ³�,B C�

l�b

and

, � , ²� � � � B³Á , � , ²� � � � B³À ²À��³� � � �� � ²�c�³ �� i i �

52

Let and and correspond as before to and (see¸� ¹ ¸� ¹ Á � �� � � � ���i � i � i �

����B ����B� < <

4.19). By the lemma,

8 � � b ¬ � À ²À��³�

� �

��

�

� � c��

¦�: ;6 7ln

��

c %��

Let

� ~ � c �  � ~ � c � Â� � � � � ��Á� � � iÁ� i i

� �

� ~ �  � ~ � ² � �Á � ~ �Á �Á �óÀ ²À��³�

� �

��

� � i i

�~� �~�

� �

� ��

v v� � �

The d.f. of is�����

- ²%³ ~ � c � � c � ²À��³�b�

¦�

c %

�

�

� �

��

% ��

��

�

�

> ?­

and thus

> � � ¬ � c � À ²À��³� ��

� �� � c %

¦��

���

We subdivide as in Theorem 4.3 into and with and defined as< < <� � � � �i i0 i00 0 00: :

before. Since

�, c - ²,³

- ²,³�Á

�

�­,¦�

for there is an such that if , then� � � � �h � � �Z

B C�, c - ², ³~ �Á

� ��

Z Z� �i0 i0

Z�

since

, ~ , ¢ � � À- ², ³

�

�

�i0 Z

Z

inf J K�

Thus if is sufficiently small and is fixed� �

, � ²� � � � ³À ²À�³� ��i i0< h

Since and by (5.17), there exist and such that if - ²�³ ~ � , � � 3 � � , � , ÁZ4 4

�, c 3, � - ²,³ � �,À ²À�³� �

Let , and be sufficiently small that equation (5.25) holds for some and� � ²�³ � � �h �

53

²��³ , � , . Then��i

4

, ~ , ~ , ¢ � � À- ², ³

� �

�

� �i i0 Z

Z

infJ K�

Together with (5.26) this implies

, ~ b6² ³ ² ¦ �³À ²À��³�

���i ��

� �

Thus, as in (5.23),

- ²%³ ~ � c � � c � À ²À��³�� �

��

c b� b6² ³

¦�

c %�

�

� �

�i

% ��

� ��

> ?6 7­

Let denote the exponential distribution with distribution function and: �² ³ � c �c %�

: : : :� � �� � i iÃi À� � �

� �: ; : ; : ;Then by (5.24) and (5.28)

> � � ¬  > � � ¬ ²� � ³À ²À� ³� � � ���

� � � i i �

� ¦� ¦��

v v v v� : � : h��

Let

Q� ��Á� �Á�~ � c Á

�² ³�

�

�: ;� ~ � c � Á � ~ � c � Á� � � �

� �

�Á� � � iÁ� i i

� �

v v

v v

and

8 ~ � c Á 8 ~ � c À ²À��³�² ³ �² ³

� � � ��Á� �Á� iÁ� iÁ�

v v v v

� �� �

� �: ; : ;By (5.20)

- ²%³ � - ²%³ ² � �Á % � Á � ~ �Á �Áà ³Á� �� �²�c�³i

�� � l

and hence

- ²%³ � - ²%³ ²� ~ � b � Á ²À��³� �� �iÁ� �Á�

v v )

so that

54

- ²%³ � - ²%³ ²� ~ � b �³À ²À��³8 8� �

iÁ� �Á� v v

Let

8 ~ � c À ²À��³�

� �

��

i i ��

��: ;e 6 7eln

Thus by Lemma 5.1

lim sup lim sup lim sup� � �¦� ¦� ¦�

8 >8

+8

- ²%³ ~ - i- ²%³ ~ - i- ²%³�

�� �

i i

� iÁ��

iÁ�v v v

� - i- ²%³ ~ - i- ²%³Á ²À��³lim sup lim sup� �¦� ¦�

+8 8>

��Á� �Á��Á�� ��

v v v

~ - ²%³ ²% � � � ~ � b �³Álim sup�¦�

8��Á� l h, j ,

where we have also used (5.29), (5.31), (5.32), Lemma 4.2, and (5.30).

We have , and thus8 i> ~ 8� ���Á� � �

�

) ) ) � l h8

> 8�

� ��Á� �� � v²!³ ²!³ ~ ²!³ ² � �Á ! � Á � � ³À ²À�³

By (5.21) and (5.24),

) ­ ! ) ­ l h� �8 >

c�

� �� �

�²!³ � c  ²!³ � c ²! � � �

��! ��!� �¦ � ¦ �: ; : ; , ),

the right sides being characteristic functions of and , respectively. Thus� �c�c %��

:6 7��

) ­ ! )� �8 ¦� �

�

��Á�²!³ � c � c � ²!³À ²À�³

��! ��!

�� : ; : ;Since is continuous at 0,)�

�² h ³

- - ²%³8 ¦� �

�

��Á� ­

�

where

�cB

B�%!

� �� �� �- ²%³ ~ ²!³Á)

and

55

lim sup�¦�

8 ��- ²%³ � - ²%³ ²% � Á � � ³À ²À��³

��Á� l h

Thus by (5.34) and (5.37)

lim sup�¦�

8 ��- ²%³ � - ²%³ ²� � ³À ²À��³

�i h

On the other hand, by (5.36),

- ²%³ � À��

�¦B

c�­c %��

Since the left side of (5.38) is independent of ,�

lim sup�¦�

8c�- ²%³ � � À ²À� ³

�

�

i

c %�

By (5.20),

- ²%³ � - ²%³ ² � �Á % � Á � � ³Á ²À��³� �� ���

�i � l h

so that

- ²%³ � - ²%³À ²À��³8 8� �� i

Combining (5.21), (5.41) and (5.39) yields

� ~ - ²%³ � - ²%³ � - ²%³ � �c� c�

¦� ¦�8 8 8

¦�

c % c %� �� i i

� �

� � �lim lim inf lim sup� � �

so that

- ²%³ � ²% � ³ ²À��³8¦�

c��

�

i

c %�­ l�

and

8 ¬ � À ²À��³�i c�c %�

�

We now form

� ~ � b � Á ²À��³� � �i i0 i00

where and are defined as in (4.25). By arguments identical to those following� �� �i0 i00

equation (4.26), is identically distributed with the occupation number� � Á� ��i0

corresponding to . We have analogously (by Theorem 4.3)- �

�

56

8 ~ � c ¬ 5²�Á �³Á ²À�³² ³ �

�� ��

i00 i00i00

¦�

� � �

�l : ;where

� ~ �²�, c - ²,³³Á�

� c �i00

�

B

,� �

�

� ��

�² ³ ~ À ²À�³

O On ln

By (5.33),

8 ~ � c c b 8 À ²À��³� �

� ² ³

�� � �i i0 i00

i00

��� � �� �

�� �: : g : ;g ;; lln

Hence by (5.45) and (5.43),

� ��� �

�: : g : ;g ;;� c c ¬ � À ²À��³� �

���

�ln

i00

¦�

c�c %��

Let

� ² ³ � � c � À� ���� �

As in (4.33),

n : ;� �

� ��

# ² ³� ² ³ c ¬ 5²�Á �³Á ²À� ³

� ² ³�

��

¦��

�

with and defined as before. Thus,� ² ³ # ² ³� �� �

� � � �� �

� �� �: ; : : g : ;g ;;� c ~ � ² ³ c c�² ³ � �

�� �

� lni00

b � ² ³ c b c À ²À�³� � �² ³

�� � �

� �� �

� �: g : ;g ;�i00

ln

By calculation,

� c c�² ³ ~ � ² ³Â ²À�³�

�i00 �

��

��� �g : ;gln

hence the second term in (5.50) converges to . By (5.48) and (5.50),��

57

9 ~ � c ¬ � À ²À�³� �² ³

² ³� �

�� � �

�: ; ¦�

c�c%��

II. We eliminate here the restriction on . As in Theorem 4.3, let have- ²�³ � :ZZ ��Á� �

spectral function such that is monotone increasing in- ² h ³ ²�³ - ²,³ c - ²,³ ,Á ²��³� �

²- ³ ²�³ ~ � ²���³ ²- ³ ²�³ ~ c � � � - ² h ³� Z � ZZ i, and . Let be defined by (4.37). Weemploy the definitions between (4.37) and (4.40). Equation (4.38) holds here also, andyields

- ²%³ � - ²%³ ²% � Á � �³Á ²À�³9 9� �i l �

where

9 ~ � c Á 9 ~ � c À�² ³ �² ³

� � � �i i� �

� �

� �: ; : ;We use arguments like those in (5.26-27) to conclude

, ~ b6² ³Á , ~ b6² ³À� �

� �� ��

� i ��

� �� �

Thus, as in (5.27-29),

> � � ¬ Á > � � ¬ Á ²À�³� � � �� �� � ¦� ¦�

� i i �

� � v v v v� : � :

where and are defined as in (5.22). As in (5.34),� �� �� �

i v v

lim inf lim inf� �¦� ¦�

9 9- ²%³ � - ²%³Á ²À³� �i �

where

9 ~ � c Á � ~ � c � À�² ³

� � � � �� � �

���

�: ;In (5.55), we have used Lemma 5.1, and

- ²%³ � - ²%³ ²� � Á � ~ � b �³Á ²À³9 9� �

iÁ� � v v h

where

9 ~ � c � c� �² ³

² ³� ��

�

�

v

v� � �

�: ;and is defined similarly. By Lemma 5.1 and (5.55)9�

iÁ� v

58

lim inf lim inf lim inf� � �¦� ¦� ¦�

9 > > 99- i- ²%³ � - i- ²%³ ~ - ²%³ ²À�³� � � ��i

� ��

where

> � � ² � �Á � � ³À� �� �� � h

Define and as in (4.25), with respect to the decomposition� �� �i0 i00

< < <� � �i i0 i00~ r Á

where corresponds to the discontinuities of and to those of< <� ��i0 i00- ²,³> ?�

> ?-²,³c- ²,³�

�À By Theorem 4.3,

� � �

�

² ³ �

�� c ¬ 5²�Á �³Á ²À�³l : ;�

�

i00i00

¦�

with as in (4.42). The measure satisfies the hypotheses of Part I, so that�i00 ��

��

�: ;� c ¬ � Á ²À ³� ² ³

��

i0 c�i0

¦�

c%��

where

� ² ³ ~ � b Á � ~ �²- ²,³ c �,³À ²À�³� �

� c �i0 i0 i0 ��

�

B

,�

�

e 6 7e �ln ��

�

Combining (5.58) and (5.59) yields

9 ~ � c b � c b� ² ³ À ²À�³� ² ³ �² ³

� � �i i0 i00 i0

i0

� � �� �

� �: ; : ;We have

�² ³ c� ² ³ ~ �� �i0 i00

so that by (5.58), the second term on the right of (5.61) converges to , and��

9 ¬ � À ²À�³��

i c�

¦�

c%��

By Lemma 5.1, (5.57), (5.53), and (5.62),

� i- ~ - i- ²%³ � - ²%³ � - ²%³ ²À�³c�

¦� ¦�9 > 9 9

¦�

c%�

��

i�

�

� � � � �: � � �6 7 lim inf lim inf lim sup

59

� - ²%³ ~ � Àlim�¦�

9c�

�

�

i

c%�

The limit yields� ¦ B

- ²%³ - ²%³ ²À�³:

�6 7���

�­�¦B

so that

9 ¬ � À�c�c

%��

III. We finally consider general for which is not identically 1. We have� ��

� ~ �²�³� b ²� c �²�³� ³À� � � �

By Theorem 3.1,

� ��

² ³ � c �²�³� c ¬ 5²�Á # ³Á ²À³�: ;� �

�

�

¦��

with

� ~ � Á # ~ � À�²,³ c �²�³ ²�²,³ c �²�³³ �

� c � ²� c �³� �

� �

B B

, , �

� ,� �� �

�

� �

Hence

� � �� �

� � �: ; : ; : ;� c ~ ²� c �²�³� ³ c b �²�³� c ¬ � Á�² ³ � �² ³ c�

� � � ��

� �

¦�

c�c

%�²�³��

²À³

completing the proof. Ê

60

Chapter 6

Physical Applications

In this chapter we examine particle number and energy distributions in certaincanonical ensembles. The means of energy distributions provide generalized Plancklaws; the classical "blackbody curve" will be shown to apply in several situations.Normality of observables is however not guaranteed, and the extreme value distributionarises in one-dimensional systems.

To simplify the discussion, we consider only particles with chemical potential 0(whose energy of creation has infimum 0), e.g., photons and neutrinos. We study these inMinkowski space and Einstein space (a spatial compactification of Minkowski space; seeSegal [Se2]), two versions of reference space. The canonical relativistic single particleHamiltonian in its scalar approximation will be used in both cases.

We study Minkowski space ensembles in arbitrary dimension. The Hamiltonian iscontinuous, and a net of discrete approximations based on localization in space will beintroduced. Einstein space will be considered in its two- as well as its (physical) four-dimensional version. The approach in these cases can be used to establish photon numberand energy distributions in a large class of Riemannian geometries, once the relevantwave equations are solved.

§6.1 The Spectral Measure: An Example in Schrödinger Theory

We begin with remarks about obtaining the spectral measure for an operator with� (continuous spectrum, acting in Minkowski space. An illustrative example of such aninfinite volume limit is in [Si].

Let be an operator on . Let denote the characteristic function of/ 3 ² ³� �9l �

) ~ ¸% ¢ O%O � 9¹ � � *9 9 �B and be its volume. Suppose that for all �

� � �²�³ � ² �²/³³lim9¦B

9c�

9Tr

exists. This defines a positive linear functional on , and there is a Borel measure * ��B �

defined by

� � � �²�³ ~ �² ³� ² ³À�This measure is the . It is the spectral measure associated with an infinitedensity of statesvolume limit, and the term is reserved for situations involving Schrödinger operators.

If is a Schrödinger operator (relevant in our context for the statistical mechanics of/non-relativistic particles), we give a criterion for existence of a density of states [Si].Assume the dimension of space is greater than 2, and define the space of potentials�

61

2 ~ = ¢ ¦ O% c &O O= ²&³O�& ~ � À�� c�c�

¦� % O%c&O�J B C K�l l : lim sup

� �

The spaces are more relevant to properties of Schrödinger operators than2 c b =� "are or local spaces. We have3 3� �

Theorem 6.1: : Let act on , with . The density(see [Si]) / ~ c b = ²%³ = � 2" l��

of states exists if and only if

B � �� 99¦B

9c� c!/²!³ ~ ² � ³lim Tr

exists.This density of states is the physically appropriate spectral measure for statistical

mechanics.

§6.2 The Spectral Approximation Theorem

Let , and be a spectral measure with spectral function ; let � � . � : -² h ³ (� � �� �ÁZ

and be the -discrete operator and occupation number r.v.'s corresponding to . Let���Z � �

¸( ¹ �� � � be a net of discrete operators with correponding occupation numbers , and�

� ~ �², ³� Á� � ���~�

B

� � (6.1)

with defined similarly. We now ask, How similar must the spectra of and be in� ( (� ��Z Z

order that and coincide asymptotically in law?� �� �Z

Let be the spectrum of the cardinality of , and< <� � � � �~ ¸, ¹ ( Á 5 ²�Á �³ q ´�Á �µ� �~�B

� ²�Á �³ � �², ³ ², ³�� � � ���~�

B

� � ��

be the truncation of with the characteristic function of . We make similar� ²,³ ´�Á �µ�, �

definitions for , and , with respect to . We will assume that there are positive<� � � �Z Z Z ZÁ 5 � (

extended real functions and such that, ² ³ , ² ³� �� �

²�³ �L �

L

²� ²�Á, ² ³³³

²� ³�

�

Z�Z ­

�¦ �

²��³ �L �

L

²� ²�Á, ² ³³³

²� ³�

�

Z�Z ­

�¦ �

²���³ �sup, ² ³�,�, ² ³� �� �

e e5 ²�Á,³

5 ²�Á,³�

�Z ­

�¦ �

²�#³ 5 ²�Á ,³ � 4 5 ²�Á4 ,³ ²� � , � , ² ³³� �� � �Z �

62

for some , independent of .4 Á 4 � �� � � We omit the proof of

Theorem 6.2: Under these assumptions, if then and have the same- ²�³ ~ � � �Z Z� �

asymptotic law i.e.,

9 ~ � c ²À�³� �

²� ³�

��l : ;L �Z

Z

and

9 ~ � c� �² ³

²� ³�

��

Z l : ;L

�

�

have identical (normal) asymptotic distributions, where

� ~ � Á�²,³

� c �Z

�

B

,� �� ²À�³

and

�² ³ c� ~ �²�³ ² ¦ �³À� �Z

Corollary 6.2.1: Conditions and may be replaced by the generally stronger²�³ ²��³ones

²� ³ ²� ²�Á , ² ³³³ ¦ �Z Z��L ��

²�� ³ ²� ², ² ³Á B³³ ¦ �ÀZ Z��L ��

§6.3 Observable Distributions in Minkowski Space

We now construct the canonical single particle Hilbert space for -dimensional� b �Minkowski space; the constructions on other spaces are made similarly. All particles willbe treated in scalar approximations, so fields will be approximated by scalar fields, andspins by spin 0. Distributions will be studied in the frame in which expected angular andlinear momentum vanishes.

We let dimensions correspond to position�

²% Á % ÁÃ Á % ³ ~� � � x

and one to time . Total space-time coordinates are% ~ !�

63

²% Á % Áà Á % ³ ~ %Â� � � ˜

the speed of light is 1 here.The field of a single Lorentz and translation invariant non-self-interacting scalar�² h ³

particle satisfies

"� � �²%³ c ²%³ ~ � ²%³Á ²À�³˜ ˜ ˜!!�

where is mass, subscripts denote differentiation, and�

" ~ b bà b ÀC C C

C% C% C%

� � �

� �� �

��

Since by assumption, we have .˜ ˜� ~ � ²%³ ~ ²%³"� �!!

The single particle space consisting of solutions of (6.4) is formulated most easily>in terms of Fourier transforms. We have if is real-valued and� > �² h ³ � ² h ³

� �²%³ ~ ²� ³ � - ²�³�%Á ²À³˜ ˜˜c ��h%��

�b��l

˜ ˜

where

� ~ ²� Á � Áà Á � ³Á �� ~ �� �� Ã�� ²À³˜ ˜� � � � � �

and

� h % ~ � % c � % À˜ ˜ � � � �

�~�

��Above, is a distribution-² h ³

- ²�³�� ~ ¸ ²� c O O³�² ³ b ²� b O O³�² c ¹O �˜ ˜ ) | ,� �� �c�k k k k k k

where

�² ³� 3 ² ³Â ²À�³

k

kl � �l

k k k~ ²� Áà Á � ³Á O O ~ � Á � ~ �� �� Ã�� Á ²À�³� � � � �

�~�

�

��: ;�

��

�² h ³ �² h ³ �² h ³ denotes the Dirac delta distribution, and is the complex conjugate of .The inner product of (we use and interchangeably) is� ² h ³Á � ² h ³ � ² h ³ �² h ³� � > �

º� Á � » ~ � À ²À ³� ² ³� ² ³

O O� �

� ��l�

k kk

k

Since (6.5) is hyperbolic, solutions correspond to Cauchy data; we have

64

�² ³ ~ O O � ²�Á ³� c � � ²�Á ³� ²À��³�

�²� ³k k x x x x

�� ��

� � �J K� �

l l

� h �!

k x k xh .

The state space thus consists of admissible Cauchy data, with time evolution dictated by(6.6). Since solutions are real, the most convenient representation of , which is� >² h ³complex, is as Lebesgue measurable functions on satisfying (6.7), with inner�² h ³ l�

product (6.9). In this representation the Hamiltonian is multiplication by ( ~ O OÀ� C� C!

kThe operator has continuous spectrum, necessitating a discrete approximation in the(

formation of a density operator. To this end, we localize in space, replacing with thel�

torus . Asymptotic distributions are largely independent of the compact manifold used;n�

the sphere will prove to give the same asymptotics.

Let be the single particle Hamiltonian on the Torus of volume , with(� n� ²� ³�

�

�

spectrum

< � t� �~ � ¢ ²� Áà Á � ³ � ² ³ � ¸, ¹ ²À��³J : ; K��~�

�

� �~�� b � B

� � �

��

.

(We neglect the 0 eigenvalue, which is without physical consequence.) The particlenumber and energy r.v.'s corresponding to in its canonical ensemble (at given inverse(�

temperature) will be

� ²�Á �³ ~ � Á ²�Á �³ ~ , � À� � � � �� ���, �� ��, ��

� � �

� �� �

"

Definition 6.3: 2 ² h ³� denotes the modified Bessel function of the second kind, of order1.

Let be the -discrete operator corresponding to the spectral measure with spectral(�Z � �

function

-²,³ ~ ², � �³Á ²À��³,�

!

��

�

�b��6 7

which is the volume of an -dimensional sphere of radius .� ,

Lemma 6.4: Let be the number of non-zero lattice points within a sphere of radius� ²%³�

% � in dimensions. Then

There is a number such that �³ 4 � ²%³ � 4 %� � ��

65

�³ For each �

� ²%³

%¦ �À ²À��³

��b��

�

!

�

6 7��

Proof: Statement clearly holds for large and small, and both sides are bounded for²�³ %intermediate values. Assertion states that the ratio of the volume of a sphere to the²�³number of its lattice points converges to 1. Ê

For brevity and convenience we define the following real functions, which arefundamentally related to asymptotic means and variances in canonical ensembles onRiemannian manifolds. They are quantities determined by "global" properties of theappropriate spectral measure , and thus arise in those less singular situations in which�asymptotics are determined by the totality of rather than its behavior near 0.�

Definition 6.5: For we define� ~ �Á �Á �ÁÃ

� ²�Á �³ ~ �,Á # ²�Á �³ ~ �,À, , �

� f � ²� f �³� �f f

� �

� ��c� �c� ,

, , �� �� �

�

The Riemann zeta function is denoted by , and² h ³

��

!�

�b��

~ À�

��

6 7In the (important) study of total particle number and energy, we will consider# ~ # ²�ÁB³ � ~ � ²�ÁB³� � � �f f f f and , which can be calculated explicitly:

� ~ ²� ~ �Á �Á �Áà ³²�³ ²�³

# ~ ²� ~ �Á �Á Áà ³²�³ ²� c �³

� ~²�³ ²�³²� c � ³  ²� � �³

² �³  ²� ~ �³

# ~

²�³ ²� c �³²� c � ³  ²� �

�c

�

�c

�

�b

�c� c�

c�

�b

�c� c�

!

�

!

�

! �

�

! �

H~��

ln

�³

� ²� ~ �³

²� ³  ²� ~ �³

�

�

c�

c�

ln .

Theorem 6.6: Under symmetric statistics,

66

(1) If or if � � � � � �Á

l : ;� ��

�� ²�Á �³ c ¬ 5²�Á # ²�Á �³³ ²À��³

� ²�Á �³ b �²�³�

�

� �c

¦�� �

c .

(2) If and ,� ~ �Á � � � � ~ �

n : ; : ;� � �

� � �O O� ²�Á �³ c ¬ 5 �Á ²À�³

� ²�Á �³ b �²�³

ln ��

� �c

¦� �.

(3) If and ,� ~ �Á � � � � ~ �

� ���� ��: ;� ²�Á �³ c ²� c � ³ c O ² ³O ¬ �� 2 ²�� ³À ²À�³� �

��

�ln lnc � c c

¦��

� �% %� �

Note that (6.16) is entirely free of the global parameters of Definition 6.5.

Proof: We first show the hypotheses of Theorem 6.2 to be satisfied in cases (1) and (2).Let and correspond as before to and , and5 5 ( (� �� �

Z Z

, ² ³ ~ O O Á , ² ³ � BÀ� �� � � �l ln��

In case (1) (if ), the lowest eigenvalue of is� ~ � (�Z

, ~ 6² ³Á��Z �

��

while

5 ²�Á, ² ³³ ~ 6 ~ 6 O O À ²À��³, ² ³

�Z �

�

�

� ��

�: ; 6 7l ln

Hence

L � � �²� ²�Á , ² ³³³ � 6 O O À�Z c

� 6 7l�� ln (6.18)

In case (2),

L � � �²� ²�Á , ² ³³³ � 6 O O�Z c�

� 6 7l ln . (6.19)

Equations (6.18-19) prove (6.14). Lemma 6.4 gives (6.15) and (6.16); hence the first twocases follow from Theorem 6.2.

If and has spectrum� ~ � � ~ �Á (�

67

< � � � � � �� ~ ¸ Á Á � Á � Á � Á � ÁùÀ ²À��³

Hence by Theorem 5.3 and the double multiplicities, the left side of (6.15) converges tothe convolution of with itself, which is on the right. � Êc�c %�

If , the distributions are explicitly� ~ � ²�ÁB³� �

l : ; : ;6 7 6 7� � �

� ! � � !� c ¬ 5 �Á ²À��³

�[ ²�³ b �²�³ �[ ²� c �³�

�

� �� �

� ��b� �b�� �

¦�

if � � �Â

n : ; : ;� � �

� � � �O O �� c ¬ 5 �Á

b �²�³

ln ��

�

� �¦�

if and� ~ �Â

� ����: ; 6 7� c O ² ³O ¬ �� 2 ���

��

ln¦�

�

c % c %� �� �

if .� ~ �For , let represent the total energy of bosons in the above� � � � � � B ²�Á �³"�

ensemble whose energies lie between and .� �

Theorem 6.7: In a boson ensemble in -dimensional Minkowski space, the� b �asymptotic energy distribution is given by

l : ;� " ��

��

�²�Á �³ c ¬ 5²�Á # ²�Á �³³À ²À��³

� ²�Á �³ b �²�³� �b�c

¦�� �b�

c

Proof: The hypotheses of Theorem 6.2 can be verified as above for ; the result"�²�Á �³then follows from Theorem 3.14 and (6.12). Ê

Definition 6.8: If denotes a symmetric or antisymmetric energy r.v.,"�²�Á �³

:; "

; "�

�

�

²�Á �³ ~ ²À��³² ²�Á �³³

² ²�ÁB³³

is the corresponding to . If the limitenergy distribution function "�²�Á �³

:; "

; "²�Á �³ ~ Á ²À��³

² ²�Á �³³

² ²�ÁB³³lim�

�

�¦�

exists, it is the of the net. If is absolutelyasymptotic energy distribution :²�Á ,³continuous,

68

�²,³ ~ ²�Á ,³�

�,:

is the asymptotic energy density of the net.

Theorem 6.7 gives the asymptotics of if , then� " " "² ²�Á �³³ ¢ ~ ²�ÁB³� � �

l : ;6 7� " �

�� !�

�c ¬ 5²�Á #³Á

��[ ²� b �³ b �²�³��

�b� �b��

¦�

where

# ~ À ²À�³�²� b �³[ ²� b �³ �

� !

��

�b� �b��6 7

Corollary 6.9: The asymptotic energy density of the Minkowski -space boson� b �canonical ensemble is

� ²,³ ~ ², � �³À ²À�³,

²� c �³�[ ²� b �³)

� �b�

,

�

�

We now find the asymptotic fermion distributions corresponding to .(�

Theorem 6.10: Let be the fermion number in the Minkowski space canonical� ²�Á �³�

ensemble. Then

l : ;� ��

�� ²�Á �³ c ¬ 5²�Á # ²�Á �³³À ²À��³

� ²�Á �³ b �²�³�

�

� �b

¦�� �

b

Similarly, for fermion energy,

l : ;� " ��

��

�²�Á �³ c ¬ 5²�Á # ²�Á �³³À ²À��³

� ²�Á �³ b �²�³� �b�b

¦�� �b�

b

Hence if " "� �~ ²�ÁB³

l : ;6 7� " �

�� !�

�c ¬ 5²�Á #³Á ²À� ³

�[� ²� b �³ ²� c � ³ b �²�³��

c�

�b� �b��

¦�

with

69

# ~ À ²À��³�²� b �³[ ²� b �³ ²� c � ³ �

� !

��

c�

�b� �b��6 7

Corollary 6.11: The fermion ensemble in Minkowski -space has asymptotic energy� b �density

� ²,³ ~ À ²À��³,

²� b �³�[ ²� b �³²� c � ³-

� �b�

, c�

�

�

§6.4 Distributions in Einstein Space

We now consider distributions in two- and four-dimensional Einstein space,< ~ : d < ~ : d (� � � �l l and . In four dimensions the single particle Hamiltonian �

is the square root of a constant perturbation of the Laplace-Beltrami operator for whichthe wave equation satisfies Huygens' principle.

We first consider Einstein 2-space, with

� ~ Á ²À��³�

9

where is the radius of the spatial portion . Clearly has spectrum (6.20), and the9 : (��

9 ¦ B asymptotics follow directly from Theorems 6.6-10. Since the discrete pre-asymptotic operator is of more direct interest here, we will(�

analyze associated distributions more carefully. We define two combinatorial functions.

Definition 6.12: If and , then� � � � � � BÁ � � tb

� ²�Á �³ � ²�²�³ b �³ ²À��³�²�³

�²h³�7 ²�Á�³ �����

� ��

where

7 ²�Á �³ ~ �² h ³ � ¢ ��²�³ ~ � Á ²À��³� ´�Á�µ

�����J K�D

and is the set of all functions . Define as theD t t´�Á�µb b

�²�³

�² h ³ ¢ q ´�Á �µ ¦ � ²�Á �³

number of ways of expressing as a sum of integers in the interval , no single� � ´�Á �µtb

integer being used more than .twice

We present the following without proof.

70

Theorem 6.13: If and is the boson energy in Einstein 2-space� � � � � � B ²�Á �³"�

at inverse temperature , then is concentrated on� "� � ²�Á �³�

�t � tb b~ ¸ � ¢ � � ¹Á ²À�³

and

F " � t² ²�Á �³ ~ �³ ~ ²� � ³Á ²À�³� ²�Á �³�

2�

��� b²�³ c �

)

where

2 ~ ²� c � ³ À ²À��³)

�~�

Bc � c�� ��

If represents fermion energy, then is replaced by , and by"� � � 2� �²�³ ²�³

)

2 ~ ²� b � ³ À ²À��³-

�~�

Bc � �� ��

The limits of the two-dimensional distributions are specializations of the 9 ¦ B � ~ �cases in §6.1.

Proposition 6.14: As , the asymptotic energy densities of bosons and��~ 9 ¦ B

fermions in Einstein 2-space are

� ²,³ ~  � ²,³ ~ À, ��,

²� c �³ ²� b �³) -

� �

, � , �� �

� �

� �� �

We now consider distributions in Einstein 4-space . The Hamiltonian acts as: d (� l �

� C� C�

(where is time) on the Hilbert space of solutions of the invariant non-self-�

interacting scalar wave equation. It has spectrum having� � h �²( ³ ~ ¸� ¢ � � ¹Á ��

multiplicity .��

Proposition 6.15: If then and are trace class.� � � � �c � ²( ³ c � ²( ³� ! � !) -� �

Proof: By Proposition 1.4 it suffices to show is trace class; we have�c (� �

tr � ~ � � ~ � � BÀ Ê ²À� ³� b �

²� c �³c ( � c �

�~�

B

�� �� ��

��

��� �

Note that

71

2 ~ � ~ ²� c � ³ ²À��³

2 ~ � ~ ²� b � ³ À

)c � ²( ³ c � c�

�~�

B

-c � ²( ³ � �

�~�

B

tr

tr

� ! ��

� ! ��

)�

-�

�

�

��

For completeness we derive the exact distribution of energy in terms of the followinginteger-valued combinatorial functions.

Definition 6.16: If and , let� � � � � � B � � tb

� ²�Á �³ ~ Á ²À��³�²�³ b � c �

� c ��²� ³

�²h³�9 ²�Á�³ �����

�

�

�

�

� �: : ;;where

9 ²�Á �³ ~ �² h ³ � ¢ �²�³ ~ � Á� ´�Á�µ

�����J K�D

with as in Definition 6.12, and denoting choose Let be theD´�Á�µ �²� ³6 7�

�� �À � ²�Á �³

�

number of ways (without regard to order) of expressing as a sum of integers in� � tb

´�Á �µ � �, each integer being used no more than times.�

Theorem 6.17: The Bose energy in Einstein 4-space at inverse temperature"�²�Á �³� �t� � Á, is concentrated on andb

F " � t² ²�Á �³ ~ �³ ~ ²� � ³Â ²À��³� ²�Á �³�

2�

��� b²� ³ c �

)

�

� � � 2 2 Àn the Fermi case is replaced by , and by � �²� ³ ²� ³

) -

� �

We now consider asymptotics of Einstein-space distributions. Denote by the -(�Z �

discrete operator corresponding to the spectral measure for which . Let� -²,³ ~ ,

�

5 ²�Á,³ ( ³ q ²�Á ,³ 5 ²�Á ,³� � � be the cardinality of ( , with the analogous definition for � Z

relative to .(�Z

Lemma 6.18: If and , then � � � , � � 5 ²�Á ,³ � 5 ²�Á �,³À� ��Z

72

Proof: We have

5 ²�Á �,³ ~ Á�,

���Z

�

�B C�

while

5 ²�Á,³ ~ � ~ b � � b � À ²À��³� , , ,

� � B C:B C ;: B C ;

�~�

�

> ?,�

� � �

We have

B C B C B C B C, , , �,� � � ² � �Á , � �³Á ²À��³

�� � � ��

� ��

�

and equations (6.42-43) imply the result. Ê

Lemma 6.19: Let be the boson number corresponding to in Einstein 4-� ²�Á ,³ (� �

space. Then

� L � ­�

¦�²� ²�Á ³³ � ²À�³�

�l

and

sup,�

Z ¦�l�

�

��g g5 ²�Á,³

5 ²�Á ,³c � �À ²À�³

�

­

Proof: We have

L ���

��

� �

�

²� ²�Á ³³ ~ � � ²À��³� �

²� c �³ ² ²�³ ³

� % �%�

² ³

~ Á�

�

��

��

� �

� �

��

�

�

� ��

� ��

�

�

�

��

�

Z

�~� �~�

²�³

²�³ � �

��

c

c

�

l � ��

> ?l< ?

l

ll

l

�

� � �

� �

from which (6.45) follows. Equation (6.46) is implied by

5 ²�Á,³ ~ ²À��³,

��Z

�

�B C�

and (6.43).

73

Lemma 6.20: If , then� � � � � � B

5 ²�Á �³

5 ²�Á �³�Á

�

��

�Z ¦�

­

and

� L ­� Z

¦B²� ²,ÁB³³ � ²À� ³� �

uniformly in .�

Proof: The first assertion follows from (6.46). The second follows from the existence offixed such that�i iÁ , � �

5 ²,Á, ³ � �5 Á , ² � Á , � , � , � B³Á,

�� �

Z Z i i� ��: ; � �

and the fact that

� L ­� Z

,¦B²� ²,ÁB³³ � ²À�³��

uniformly in . � Ê

Theorem 6.21: The boson number in Einstein 4-space satisfies

l : ;� ­�

��

� ¦�� ²�Á �³ c 5²�Á #³Á

�² ³�

�

where

�² ³ ~ �, b �²�³ ² ¦ �³ ²À�³� ,

� � c �� ��

�

� �

,�

���

# ~ �,À� , �

� ²� c �³��

� � ,

, �

�

�

If represents fermion number, each “ ” is replaced by “ ”.� c b�

Proof: This follows in the Bose case from Lemmas 6.18-20, Theorem 6.1 and itsCorollary, and Theorem 3.14. Ê

74

Theorem 6.22: The Einstein 4-space asymptotic boson and fermion number and energydensities are identical to those in Minkowski 4-space. Precisely, equations (6.14), (6.22),and (6.27-28) still hold.

Corollary 6.23: The asymptotic energy densities of Bose and Fermi ensembles inEinstein 4-space are

� ²,³ ~�,

²� c �³)

� �

, ��

�

��

� ²,³ ~ À���,

�²� b �³-

� �

, ��

�

��

§6.5 Physical Discussion

The requirements of localization in physical (as well as momentum) space result inthe approximation in §6.2 of the Hamiltonian for massless scalar particles in Minkowskispace. There seems to be no direct way of discretizing without reference to .� ²(³ (!The r.v. is interpreted as boson number in an -dimensional torus of volume� ��

= ~ � � �²� ³ �²� ³

�

� �

��� �

c

�. If the expected number per unit volume is asymptotically ; the

same density occurs for Einstein 4-space (Theorem 6.21). If the mean density of� ~ �

photons in the energy interval is asymptotically , and thus large (and´�Á �µ ² ¦ �³O ² ³Oln ��

���

9-dependent) in Einstein space and infinite in Minkowski space. Since asymptoticdensity is independent of , the divergence clearly arises from bosons with effectively�vanishing energy; this is an effect of Bose condensation, in which large numbers ofparticles appear in the lowest energy levels. The corresponding spatial energy density is,however, finite according to Theorem 6.7, which gives mean density

+ ~ �,À� ,

� � c ��

� �b��

�

,

,� !

��

�

6 7 � �

This divergent density of low-energy bosons with finite corresponding energy density hasan analog in the "infrared catastrophe" of quantum electrodynamics (see [BD], §17.10), inwhich an infinite number of "soft photons" with finite total energy is emitted by anelectrical current. The extreme density is correlated with the lack of normality in photonnumbers.

The energy distribution of bosons in a Minkowski -space canonical ensemble in� b �(6.25) is the “Planck law” for such a system. An observer of scalar photons measures theproportion of photon energy in the frequency interval to be´ Á b µ� � "��

�² b ³

)�

� "�

� ²,³�, �, with Planck's constant. Analogously, (6.31) gives the

corresponding law for fermions; in an ensemble consisting of neutrinos, the proportion of

75

total neutrino energy observed in is thus . Note that the´,Á , b ,µ � ²,³�," ,

,b ,

-

"

Planck laws for Einstein 4-space coincide asymptotically with those in Minkowski space;this obviously also holds in one dimension. This indicates that the cosmic backgroundradiation expected in an approximately steady-state model of the universe is largelyindependent of the underlying manifold. The specific correspondences in this chapter area consequence of the physical identity of Minkowski space and the “ limit” of9 ¦ BEinstein space of radius .9

76

Chapter 7

The Lebesgue Integral

In this chapter we introduce a natural and useful generalization of the notions inChapter 3. Lebesgue integration of r.v.-valued functions on a measure space is themaximal completion of Riemann integration. The step from Riemann to Lebesgueintegration shifts the focus from the domain to the range of the integrated function;indeed, the ordinary Lebesgue integral is a Riemann integral of the identity function onthe range space with respect to the domain-induced measure; this viewpoint will be usedhere.

The present integration theory is in fact interpretable as a formal extension of thetheory of semi-stable stochastic processes (see [La, BDK]), with an abstract measurespace replacing time. The r.v.-valued function being integrated yields an r.v.-valued?measure defined by integral of over ; this measure is clearly aC C²(³ ~ ? (generalized stochastic process. Indeed, in the notation of this chapter, if is a¸?²!³¹!�l

collection of independent standard normal r.v.'s, then is simply Brownian�

!�°�?²!³²�!³

motion.The advantage of the present approach to that of standard integrals of distribution-

valued random functions (e.g., “white noise” ) [Che,R1,V] is that it does not require theexistence of a metric (or smooth volume element) on the underlying measure space.Precisely, the present integration theory would, on a Riemannian manifold, be equivalentto (linear) integration of an r.v.-valued distribution with covariance

; � � L � � � � � $²?² ³?² ³³ ~ ²?² ³³ ² Á ³ ² � ³ À� � � � � �

(Here, denotes the point mass at in the variable ) However, an abstract measure� � �� �Á Àspace generally has no such object.

A novel aspect of the Lebesgue integral is the use of a non-linear volume element� �²� ³. This may seem rather ad hoc; but in Chapter 8 the Lebesgue integral is shown tobe isomorphic to a linear integral over functions with range in a space of logarithms ofcharacteristic functions.

Lebesgue integration is the natural environment for detailed study of integrals ofindependent random variables; however, aside from proof of associated fundamentalswhich will comprise most of this and the next chapter, the approach here will berelatively goal-and-applications oriented. Further on, the measure space will² Á Á ³$ 8 �be the spectrum of an abelian von Neumann algebra of quantum observables.

The material in the next three chapters will be largely independent of previousmaterial, and the probabilistic content stands on its own. The proofs may be omitted on afirst reading.

77

§7.1 Definitions and Probabilistic Background

We will often deal with normalized integrals (sums) of random variables with infinitevariance; the resulting limits will depend strongly on the tails of the integrateddistributions. We recall the bare essentials of general limit theorems for sums ofindependent random variables; see [GK] for details.

In order to formulate the most general central limit results, we define the Lévy-Khinchin transform of an infinitely divisible distribution. Recall every such distribution� ) � has characteristic function , where is continuous and vanishes at The~ � ! ~ �À�

book of Loève [Lo] has a proof of

Theorem 7.1: The distribution is infinitely divisible if and only if , where � ) �~ ��

is given (uniquely) by

� �²!³ ~ � ! b � c � c �.²%³Á ²�À�³�!% � b %

� b % %� : ;cB

B�!%

� �

�

with and is a non-negative multiple of a d.f.� l� .

The pair is the of . Note this is additive, in that if² Á .³� �Lévy-Khinchin transform� � � � � �� � � � � � � � � has transform , then has transform . It can² Á . ³ ²� ~ �Á �³ i ² b Á . b. ³be shown that the transform is continuous from the space of distributions in the�topology of weak convergence, to the pairs in the topology of crossed with the² Á .³� l�

topology of weak convergence.Let be a double array of independent random¸? ¹Á � � � � BÁ � � � � ��� �

variables; is allowed.� ~ B�

Definition 7.2: The variables are if for every ? � �Á�� infinitesimal �

sup�����

���

7²O? O � ³ �À� ­� ¦ B

For any monotone functions of bounded variation, we write. ²%³Á .²%³�

. ²%³ ¬ .²%³�� if the same is true of the corresponding measures on .l

Lemma 7.3.1: Let be a sequence of independent r.v.'s. In order that ¸? ¹ ?� ��~�

B�converge weakly and order-independently, it is necessary and sufficient that

� �: ; : ;�~� �~�

B B��

� �� �

�; ;

? ?

� b? � b?Á ²�À�³

converge absolutely.

Proof: By the three-series theorem, order-independent convergence above is for any� � � equivalent to absolute convergence of the series

78

� � �� � �� � �O%O� O%O� O%O�

� � ��

� � �

�- ²%³Á %�- ²%³Á % �- ²%³Á ²�À�³

where is the d.f. of . But absolute convergence of (7.3) implies convergence of the- ?� �

first series in (7.2). Thus proving absolute convergence of the second series reduces tothe same for

� �� �: ;� �O%O� O%O�

� �� �

�

� �

% %

� b % � b %c % �- ²%³ ~ c �- ²%³À ²�À�³

Convergence of the latter follows from that of the first series in (7.2).

Conversely, if (7.2) converges absolutely, so do the first and third series of (7.3),while convergence of the middle series follows from a subtraction argument, as in (7.4).Ê

Henceforth, will denote the (cumulative) distribution function of , and all- ²%³ ?�� ��

infinite sums must converge order-independently to be well defined. Assume that{ form an infinitesimal array. The proof of the following theorem for finite is? ¹ ��� ��~�

��

in Loève [Lo]; we extend it to general .��

Theorem 7.3: Let . In order that converge weakly to a distribution ,: ~ ? : :� �� ��

�i is necessary and sufficient that!

. ¬ .Á ¦ À ²�À³� �� �

Here

��

� �

~ � b �- ²&³Á . ²%³ ~ �- ²&³Á ²�À³& &

� b & � b &� �� ��~� �~�

� �

�� �� � ��cB cB

B %

� �

�

and

� ~ &�- ²&³Á - ²%³ ~ - ²% b � ³Á�� �� �� �� ��O&O�

��

with any fixed constant. The pair is the Lévy-Khinchin transform of .� �� � ² Á.³ :

Proof: This follows using approximation by finite sums. For example, if converge:�

weakly to , the same is true of , where is finite but sufficiently large;: : ~ ? �� �i i

�~�

�

����i

using the result for and letting become infinite proves (7.5). The only difficulty: �� �i i

lies in proving the sum defining converges (order-independently) if and only if (7.6):�

does. To this end, note that if exists, the three-series theorem implies : O� O� ���~�

B�

79

converges. Letting ,�²&³ ~ &�b&�

� �� �� �

�~� �~�

B B

cB cB

B B

�� �� ��

�~�

B

cB

B

�� �� ��Z

� �

� �

²&³ �- ²&³ ~ ²& c � ³�- ²&³ ²�À�³

~ ² ²&³ c � ²& ³³�- ²&³

where is determined according to the mean value theorem. By& c � � & � &�� ��

infinitesimality of the , this sum converges if and only if ? ²&³�- ²&³�� ���~�

B

cB

B� �

converges. This (again three-series) occurs if and only if converges� �~�

B

c ���

��²&³�- ²&³

for some . The latter follows from the Lemma. The existence of the limit � � � .²%³follows similarly.

Conversely, assume the limits (7.6) exist. Then the sum

� ��~�

B

�� ��c

� b &�- ²&³�

�

converges for any , since is dominated by for . Therefore we� � �� � & c ²&³ O&O �&�b&

�

� have convergence of

� � ���: ;� �

�~� �~� �~�

B B B

�� �� �� �� �� �� ��c b�

b�

�~�

B b� c b�

c��

� b ²& c � ³�- ²&³ ~ � b � 7²O? c � O � ³ ²�À�³

b c &�- ²&³

�

�

� �

� �

��

��

b

�� ��c

�

.

Using infinitesimality of and finiteness of for (the latter� 7 ²O? O � *³ * � ��� ���

�follows from convergence of the second sum in (7.6)), the last two sums converge.

Therefore is convergent, as is . Using (7.7) and the infinitesimality� ��

�� ��cB

B� �-�

of , we conclude that converges (absolutely) as well. Similarly ¸� ¹ �-�� ��� �

cB

B� � �

cB

B &�b& �� �

�

� �- � B : Ê. Thus, by Lemma 7.2.1, exists and is order-independent.

Henceforth let

� � � �i�%

� �

�

²%³ ~ Á ²%³ ~ Á ² ³ ~ À ²�À ³% O%O � �

 O%O � �

% %

� b % � b %H

Corollary 7.3.1: In order that converge weakly, it is necessary and sufficient that:�

80

� �Z

�¦B�~� �~�

� �

�� ��cB

%

~ � Á .²%³ ~ ²&³�- ²&³ ˜lim lim� ��� �

w -

exist, where

� ~ ²&³�- ²&³Á - ²&³ ~ - ²& b � ³ ²�À��³�� �� �� �� ��cB

Bi� � ˜ .

The weak limit has Lévy-Khinchin transform , where: ² Á.²"³³�

� �~ � b ²&³�- ²&³À ˜lim�¦B

�~�

�

�� ��cB

B� ��

Proof: If converges weakly, then for some: . ²%³ ~ ²&³�- ²&³ ¬ . ²%³� � ���

cB

%

�¦B

i� �

multiple of a d.f. If are continuity points of , then. f .i i�

� � �� cB cB

c c

�� �

�

�

� �

�- ²&³ ~ �. ²&³ ²�À��³� b &

&

converges as . Similarly, converges; and� ¦ B �- ²&³��

O&O� ���

� � �� O&O� O&O�

�

� �� �� �

&

� b &�- ²&³ ~ &�. ²&³

converges as well, as does . Thus, by convergence of in (7.6)��

O&O�&

�b& �� �� � �- ²&³ �

and the identity , we have convergence of% c ~% %�b% �b%� �

�

�: ;��~�

�

�� ��O&O�

�

� b &�- ²&³ Á�

with as in the Theorem. We conclude the convergence of���

� �� �: ;�: ;�

�~� �~�

� �

�� �� �� �� ��c c� O&O�

c�

�~�

�

�� �� ��O&O�

� �

��

��

�

� b &�- ²&³ ~ � b ²& c � ³�- ²&³ ²�À��³

~ � b � �- Â

� �

�

�

the sum clearly converges to 0 as by infinitesimality of the� �~�

�

�� ��O&O�

�

� �- � ¦ B�

� � � ¦ B�� ���~�

�

, and hence converges to a limit as .��

81

On the other hand, since for converges weakly (equation (7.11)),% � �Á - ²%³� ��

� �: ;� �cB cB

c c� ci i

�� �� ��

� �

� ���

� �²% b � ³� - ²%³ ~ ²%³�-

converges, so that the sum defining converges (the sum converges by� �Z i

�

B��

��

�-

similar arguments). Let , and consider� � � c ��� �� ��

� �� ��: ;� �

� �cB cB

% %c�

�� �� �� ��

� cB cB

%c� "c�

�� ����i Z

� �

� �

²&³�- ²& b � ³ ~ ²& b � ³�- ²&³ ²�À��³

~ ²&³�- b � ²&³�- Á

��

�� ��

where is between 0 and . Without loss of generality, we let ; then� � � ���i

�� �

� � �g g�� � �

�� �� �� ��O%O�

iO� O ~ O� c � O ~ ²&³�- ²&³ ²�À��³�

�

remains bounded as , as does . We have � ¦ B O� O O ²&³O�- �� ��i Z

cB

B��� ­

� ¦ B

uniformly in , so that the last term on the right of (7.13) vanishes as , and� � ¦ B

� �� cB

%

��i�²&³�- ²&³ ¬ . ²"³ ~ .²"³À ²�À�³˜

Conversely, if the series for and converge absolutely in and as , then it�Z .²"³ � � ¦ B

follows along similar lines that converges as , for any� �~�

�

O&O�i

��

�

�� ²&³�- ²&³ � ¦ B

� �� �À �- � ¦ BÁ � Thus, since converges as converges as well. Using� ��~� �

�i

�� ��

�

by now standard arguments, it follows that converges as ��

cB

B���²&³�- ²&³ � ¦ BÁ

as does Similarly, if is a continuity point of then� ²� b ²&³�- ²&³³À % .²%³Á�� ��cB

B�

��

cB

%���²&³�- ²&³ ¦ .²%³.

We now prove the last statement of the Corollary. We have

lim�¦B

�

�� ��cB

B�B C�� b �- ~�

82

lim�¦B

�

�� �� ��cB cB

B Bii�B : ; C� �� b �- b �- Á ²�À�³

c� �

�� �˜

since the measures and have the same weak limit (see (7.15)). By˜� �� �

�� ��� ��- �-

the mean value theorem

� �� � �� : ;�

� �cB cB cB

B B Bi i i i

�� �� �� ��Z

��

�

�� ��cB

Bi iZ

��

� � �

�

�- ~ �- c � ²& c � ²&³³�- ²�À��³

~ � ²� c ²& c � ²&³³�- ³ À

˜

Since is uniformly bounded (since is), and� � � �

cB O&O��

B i iZ�� �� ��²� c ²& c � ³³�- �-�

� � � ¦ BÀ�� ­� ¦ B

uniformly, (7.17) vanishes as Similarly, if is a continuity point�

of and.

��

�ii²%³ ~

% O%O �� O%O � ÁH

then uniformly. Thus (7.16) is given by��

cB

B ii��� ­�- �

� ¦ B

lim�¦B

�

�� �� ��cB cB

B Bi ii�B C� �� b ² c ³�- b �-� � � ˜

~ � b ² c ³�- b �- ²�À��³

~ � b �- b �-�

&

~ � b �- Á

lim

lim

lim

�¦B�

�� �� ��cB cB

B Bi ii

�¦B�

�� �� ��O&O� cB

B

�¦B�

�� ��cB

B

�B C� ��B C� �� �

� � �

�

�

˜ ˜

˜ ˜

˜

�

the last equality following from

�� �� O&O� O&b� O�

�� ��� �

� �

& &�- c �- �À˜ ˜

��

­� ¦ B

Equation (7.18), together with (7.15), completes the proof. Ê

83

§7.2 Definition of the Lebesgue Integral

Since we will study integrals of measure-valued functions, we consider metrics onspaces of probability distributions. Let be the set of probability Borel measures on : l�Áand the set of finite Borel measures. Define the Lévy metric by: �i 3

� � � � � :3 i� � � � � � �² Á ³ ~ ¸� ¢ - ²% c �³ c � � - ²%³ � - ²% b �³ b �¹Á ² Á � ³Áinf

²�À� ³where is the d.f. of . The Lévy metric is compatible with the topology of-� ��convergence in distribution on (see, e.g., [GK]). Note that can be defined for any: �3

pair of Borel measures on . This general definition will be used here.� � l� �ÁWe will also require a stronger metric which emphasizes tail properties of measures.

Let : be defined for small arguments, and nonvanishing. Define (recall (7.9))� l lb b¦

. ²%³ ~ ²&³ � À ²�À��³� &

² ³� �Á

cB

%

� � �� �� : ;

We define

� � � � � � �� � �

� � � � � �Á � � Á Á � �3

cB

B

² Á ³ ~ ². Á. ³ b ²&³�² c ³ À� &

² ³� � g : ;g�The measure is defined by�²�%³

� �²�%³²*³ ~ ²�*³Á

for a Borel set in . Since the integrand of (7.20) vanishes only at , it follows* % ~ �lthat for fixed , and are equivalent. We introduce the strengthened metric� � �3

Á� �

� � � � � �� � ��

² Á ³ ~ ² Á ³À ²�À��³� � Á � ��� ��

sup

This will be useful for our integration theory, in which tail behavior of probabilitydistributions will be crucial. Note that infinite distances under are not excluded; this is��clearly an inconsequential deviation from the standard properties of a metric. We willnow require a proposition. We define for , ,� ¢ ¦ � �l l �

² �³²%³ ~ À�²% b ³ c �²% c ³

lip² ³

�� �

�

� �

sup g g� �

�

If is a measure on , then is the total mass of .� l � �O O

Proposition 7.4: Let be absolutely continuous and of bounded variation, and� l l¢ ¦� � l � � � � � �� � � � B

3 Z�

Á ~ ² Á ³ P P Á P be finite Borel measures on . Let , and assume that ,h

84

and are finite, where denotes the derivative. Thenh hlip² ³ Z Z�

� O O²%³� �

� > ?h h hcB

B

� � B � �Z ² ³ Z

�� � � � � � � � �² c ³ � P P b � b � O O²%³P O O Àlip �

Proof: If (allowing a slight abuse of notation), then� �� �cB

% Z²%³ ~ � ²% ³g g g g� �

cB cB

B B

� � � � � �cBB� � � � � � � � � �² c ³ � O ²%³² c ³²%³O O b ² c ³²%³� ²%³

� b ² ²% b ³ b c ² ²% c ³ c ³³ O� ²%³O

� ² b � ³ b ² ²%b ³ c ²% c ³³O ²%³O�%

~ ² b � ³ b ²%³²O ²% c ³O c O ²%

� � � � � � � � �

� � � � � � � �

� � � � � � �

h h �h h h h �h h h h �

BcB

B

� �

B �Z Z

cB

B

� �

B �Z Z Z

cB

B

� b ³O³�%

~ ²P P b �P P ³ b ²%³� ²%³Á

�

� � � � �B � �Z

cB

B� �

where

� � � � ��²%³ ~ ²O ²& c ³O c O ²& b ³O³�&Á��

%Z Z

and

O ²%³O � P ²%³P À� � ���lip² ³ Z

�

Thus

k k� �cB cB

B B

� � B � �Z B

� cB� � � � � � � � � ��² c ³ � ²P P b �P P ³ b O c �� �

� P P b �P P b �P ²%³P O O Á� � � � �6 7B � � �Z ² ³ Zlip �

as desired. Ê

Remark: If is any function for which satisfies the hypotheses of� �²%³ ~ � �

�

²%³c ²%³²%³

Proposition 7.4, then a metric equivalent to obtains by replacing by in the� � ��

definition. To see this, note that by Proposition 7.4

85

g : ;g g : ;g� �� % � %² c ³�² c ³ ~ � �² c ³ � -² ² Á ³³Á

� � � �� � � � � � � � � �

cB cB

B B

� � � � � ��

where , and is a continuous increasing function vanishing at 0.� ~ -� �

�

c

Let be a -finite measure space, and be , in that² Á Á ³ ? ¢ ¦$ 8 � � $ : measurable ? ² ³ � � À ²?³c� E 8 � E : H for every -open Let be the range, and�

H � H � � � �ess²?³ ~ ¸ � ²?³ ¢ ²? ²) ² ³³³ � � D � �¹ )c�� � �, where is a -ball of radius

� HÀ ²?³ Note for future reference that gives the full picture with regard to integrationess

theory, since

� � � H¸ ¢ ?² ³ ¤ ²?³¹ ~ �À ²�À��³ess

For otherwise, there would exist an uncountable number of open balls in ,¸) ¹� ��4 :with

� �? ²) ³ ~ �Á � � ? ) � Bc� c�� �

�: ;�

(recall is -finite). Such a situation is equivalent (by collapsing each ) to a discrete$ � )�

non-atomic measure space of positive finite total measure, which does not exist.By analogy with the Lebesgue integral, we initially assume is finite and is -� �? �

bounded, i.e., the diameter of its range is bounded. Let be a sequence of at¸7 ¹� �~�B

most countable partitions of , each with elements . ForH Hess ess²?³ ¸7 ¹Á 7 ~ ²?³� �� ��

�: � : ~ ¸ ² Á ³ ¢ � :¹: � � � �, let diam sup . We assume� � �

²�³ 4²7 ³ � ² 7 ³ �� �sup�

�diam ­�¦B

²��³ �²7 ³ � ²7 ³ �À� �sup�

�� ­�¦B

Hence the mesh of vanishes both in diameter and measure . For the purposes7 ²�³ ²��³�

of this definition, as in Chapter 3, atoms of may be artificially divided into pieces � �� �

and apportioned to various , as long as 7 ²� ³ ~ ²�³À�� ��� �

Definition 7.5: A partition net satisfying and is .²�³ ²��³ infinitesimal

Let be defined for small positive , and vanish at 0. We say that� � �² ³

3 � �$ �

� �?² ³ ²� ² ³³ � ?² ³ ² ? ²7 ³³� � � � � � �lim¦B

� �c�

exists if the right side is independent of choice of and of satisfying �� � �� � �~�B� 7 ¸7 ¹ ²�³

and . This is the of with respect to .²��³ ? finite Lebesgue integral �

86

Remark: The finite Lebesgue integral does not depend on whether the partitions are7��over the range or the essential range H H²?³ ²?³Àess

By our assumption of -finiteness of , is separable. For if it were not, there� $ Hess²?³would exist an uncountable disjoint collection of balls in , with¸) ¹� :� � �²? ²) ³³ � �c�

� , contradicting the -finiteness of .

For the case of a general -finite measure and measurable function , we� � $ :? ¢ ¦disjointly partition ; we assume , diam , whereH �ess ~ 7 ²7 ³ 7 � B�

7 �7 � � �i

�

� �i c�~ ? . The partition is always assumed at most countable, which is possible sinceH �ess²?³ is separable and -finite. The general Lebesgue integral

3 3c�

�

� � ��$ H

?² ³ ²� ² ³³ � ²� ² ³³ � ? ²� ³ ²�À��³� � � � �� � � � �ess

i

�

i

? ²7 ³

is defined using finite integrals on the right, and exists if the sum on the right isindependent of the partition . The above independence condition, which may seem7difficult to test, is natural; see Theorem 8.4.

We must verify that the general Lebesgue integral coincides with the finite one ifH²?³ is finite, bounded and separable. To this end, we require

Proposition 7.6: If and the finite Lebesgue integral exists, and if: : �� ��i� ²� ³

3 �

:

: : �� �� �i� ²� ³has positive Borel measure, then exists.

:�

Proposition 7.7: If the finite integral exists and is a partition of ,3 �

: ��� � :²� ³ 7i

�

then

3

� �

3� ��:

�� � � �� � �²� ² ³³ ~ ²� ² ³³Á ²�À��³i i

�

i

7

�

order independently, where ��

i

� � �� � �~ i ià 
Proof of Proposition 7.6: We will require the fact that the finite Lebesgue integral isalways infinitely divisible (this follows from the definition and a stronger result is provedin Theorem 7.9). Hence the characteristic function of does not) �� �²!³ @ ~ ²� ³

:�

i

vanish. Let be an infinitesimal sequence of partitions of , and a sequence for7 7� �²�³ ²�³

�:

: :� � � � � �²�³ ²�³ ²�³ ²�³

� ? � 7 Á ? � 7. Let , and� � � �

? ~ ? ² ³ ²� ~ �Á �³À� � ��

�� �

²�³ ²�³� � �

If is unbounded (i.e., has a subsequence converging in law to a measure strictly less?�²�³

than 1 as ) for or , then cannot exist. Thus, there is a subsequence of� ¦ B � ~ � � @

87

¸? ¹�²�³ which converges in law to some probability distribution, and without loss of

generality we reindex so that , where has ch.f. . Thus,? ? ¬ ? ?� ��

²�³ ²�³

¦B

²�³ ²�³ ²�³)

? i? ¬ @ Â� �²�³ ²�³

hence

? i? ¬ @ Á²�³ ²�³�

or in terms of characteristic functions,

) ) ­²�³ ²�³

¦B²!³ ²!³ "²!³À ²�À�³� �

Since , this shows that converges, as does in law. By (7.25),) )²!³ £ � ²!³ ?� �²�³ ²�³

w- is independent of the choice , proving Proposition 7.6. lim�

� �¦B

²�³ ²�³? 7 Ê

Sketch of proof of Proposition 7.7: Note that terms on the right in (7.24) can be well

approximated by “Lebesgue sums” , where are elements of a�? ² ³ ¸? ¹� ��Z ZZ� ��i� �

subpartition of . Since the approximants can be made to converge order-independently7�to the left side, so can their limits

7i

���� � �²� ² ³³À Ê

In this chapter an assortment of distributions will obtain as Lebesgue integrals, as noprior constraints are placed on existence of the mean and variance of the integrand.

The Lebesgue integral is clearly linear, i.e.,

3 33� � �²? b? ³ ²� ³ ~ ? ²� ³ b ? ²� ³Á� � � �i i i� � � � � �

if and are independent for each It is also additive, i.e., the integral over a? ? � À� � � $union of disjoint sets is the sum (i.e., convolution, in the distribution picture) of, r ,� �

the integrals over and ., ,� �

Henceforth, all integrals of real-valued functions will be Lebesgue integrals. Clearly,our integral reduces to standard Lebesgue integration when is the identity, and is� �?² ³a point mass for .� $�

Theorem 7.8: Let have 0 mean a.e. , with and ?² ³ ´ µ # ~ ²? ³� O ²? ³O�� � ; � ; � $ $

� �

finite. Then the Lebesgue integral of exists, and?

3

���

$

?² ³²� ² ³³ ~ 5²�Á #³À� � �

Proof: Let and be as above, and let . We¸7 ¹ ¸ ¹ ? ~ ?² ³Á ~ ? ²7 ³� � � � � �� � � �� � � � �c�

invoke Theorem 3.6, and note that

88

�

�

; �

; �­�

�

� �

� ��

�

�

���

�

��

��

¦�� �Á

²O? O ³

² ²? ³ ³

��

��

��

since the denominator approaches the integral , while the numerator² ²? ² ³³� ² ³³ ; � � �� ��

vanishes, given that . This showssup� �¦B

� ­��

�

��

� �¦B

? ¬ 5²�Á #³À Ê� ��

���

§7.3 Basic Properties

Definition 7.9: A probability distribution is if to every there� stable � Á � � �Á � Á � Á� � � �

correspond constants and such that� � � �

- ²� % b � ³i- ²� % b � ³ ~ -²�% b �³Á� � � �

where is the d.f. of . If , can be chosen to be 0, then is a .- � Á � �� �� � scaling stable

Stable distributions, which are intimately connected to the Lebesgue integral, can becharacterized by

Theorem 7.10: : In order that be stable, it is necessary(Khinchin and Lévy, [KL]) �and sufficient that its characteristic function satisfy)

� ) � � �²!³ � ²!³ ~ � ! c �O!O � b � ²!Á ³ Á ²�À�³!

O!Oln �J K

where , , and� l � �� c � � � �Á � � � �Á � � �

�� �

�²!Á ³ ~ À

£ �

O!O ~ �H tan

ln

�

�

��

if

if

Corollary 7.10.1: In order that be scaling stable, it is necessary and sufficient that the�logarithm of its characteristic function have the form

�²!³ ~ c � O!O b �� O!O Á ²�À��³!

O!O� �

� �

where and ; if is arbitrary.� � �Á � � � �Á O� O � � O O ~ �Á � �� � � ��� � � ltan �

89

Proof: The scaling stability condition is . Sufficiency is clear,� � �6 7 6 7 6 7! ! !� � �� �

b ~

so we prove only necessity. If is stable, is given by equation (7.26). We assume� �²!³momentarily that . Using (7.26), we have� £ �

� ! b c �O!O b � b � ~� � � � !

� � � � O!O �� � �

�: ; : ;J K� � � �

�� � tan

� ! c �O!O � b � À� � !

� � O!O �� � �

�: ; : ;J K�

�tan

We conclude that or . In either case, the function fits the form²�³ ~ � ²��³ � ~ �� �(7.26). If , a similar argument shows , completing the proof. � �~ � ~ � Ê

The proof of the following lemma uses arguments similar to those in Theorem 7.3 andits Corollary, and is omitted.

Lemma 7.11.1: Let be independent r.v.'s, and converge order¸? ¹ ?� � ��

�independently. Let be an infinitesimal array of real numbers (i.e., ,¸� ¹ � ��� �� ­

� ¦ B

uniformly in ). Then � � ? ¬ �À��

�� ��¦B

We have the following characterization of Lebesgue integrals.

Theorem 7.11: In order that be the distribution of a Lebesgue integral of an r.v.-�valued function, it is necessary and sufficient that be scaling stable.�

Proof: Assume . It is easy to see that then . Let@ ~ ?² ³ ²� ² ³³ £ � ² ³ �3 � � � � � � ­

� ¦ �

: � :� �i be the essential range of and the induced measure on . Since scaling stability?

is preserved under sums, there is no loss in assuming is -bounded and finite in: �� �

measure. Let be a sequence of positive numbers. Let be a� ~ � � � � � � à 7� � � ���

partition of into an at most countable collection of sets of measure smaller than ,:�c�� �

with -diameter less than . For , let be a sub-partition of such that given���� � � �

� �� � � 7 7�

any , there is at most one element with and7 � 7 7 � 7 7 � 7� � �� �� � �� � � � � �

� �� � � ��

�i � c��� �²7 ³ £ � � � �

� (note changes with and ). For simplicity, we assume such an

element exists for all ; its measure may be 0. Recall that singletons can be� � � � 7� ���

sub-divided (along with their measures), for purposes of partitioning; if this were onlyallowed for singletons of positive measure, the following arguments would be somewhat

more technical. Fix as above, and let be constructed in such a way that � 7��

� ����6�� �b�

7 7 ? ��� � � � �� � �

� � � �7� is non-empty, containing a fixed element, denoted by , for each . For

� � � ? ~ ? � 7 ?� �� � � �� � �� � � �, choose , where is as above.

� � � � � �

90

For each with , choose . For , let7 � 7 ²7 ³ ~ � ? � 7 � � ��� �� �� ��� � � c� � �

� ��

7 ~ ¸7 � 7 ¢ ²7 ³ £ � ¹��i � i � c�

�� ��� � , and consider

: ~ ? ² ²7 ³³Á ²�À��³� �� ���i � i �

7 �7

��� �

�i

� �

which is order-independent by definition of the Lebesgue integral. Given , we� Áà Á �� �

note that since can be made sufficiently large that for �i c� �i�� �b� �b� �²7 ³ � � Á � � � � Á :

is arbitrarily small (i.e., close to the unit mass at 0). This follows by Lemma 7.11.1, andfrom the fact that at most are representative of each of the sets appears in the sum¸7 ¹� �

���

(7.28). We thus successively select so that , uniformly in� Á � Áà ²: Á ³ �� � �3 �i

�� � ­

� ¦ B

� � ´� ÁB³�b� .

We now choose our final partitions. Let

7 � 7 Á ? � ? Á 7 � 7 Á : � : ²� � ´� Á � ³³À� �� �b� �b�� �� � � � �� � i �i i i�

By the above,

: ¬ �À�i

�¦B

Thus

: � ? ²� ³ ¬ @ ~ ?² ³ ²� ² ³³À ²�À� ³� ��

7 ¤7

c�

�¦B� ��� �

i

3� � � � �

$

However, each element of may be divided exactly in half, ,7 � 7 7 ~ 7 r 7� ���i

�� ��²�³ ²�³

and we may choose independent copies of , to be contained in and? Á ? ? 7�� ��²�³ ²�³ ²�³

�� �

7�²�³; respectively. Then

� �: ; : ; 6 77 ¤7 7 ¤7

�� ��²�³ ²�³

c� c���

c� � ��¦B

�� ��� �i i

c�

? b ? ~ ²: i: ³ ¬ @ À� �

� � ²� ³� �

�

�

Thus exists, and in distribution. Similarly, for�²�³ ~ @ � @ i@ ~ �²�³@lim�

� �

� �¦�

²� ³² ³

i�

� � Á @ ~ �²�³@ �²�³ � @h l �i� for some . Thus is infinitely divisible; if is thelogarithm of its ch.f. then

� ²!³ ~ ²�²�³!³ ²� ~ �Á �Áà ³À ²�À���³� �

Thus,

� �

� �²�³²!³ ~ ! Á ²�À���³� �: ;

91

and . We define If , let , the� �� �²�³ � �²�³

�²�³ �²�³� �� � � � k²!³ ~ ! � ~ À � � � ¦ Á � �6 7 4 5

rationals. Then

�� � �²!³ ~ � ²!³ ~ ²�²� ³!³Àlim lim�¦B �¦B

� �

This holds for all . Thus if is not identically exists, is! � �Á �²� ³ � �² ³ �l � �lim�¦B

�

continuous, and . Letting , we have . Thus, if�� � � � l²!³ ~ ²�² ³!³ ² � ³ � ¦ � �²�³ ~ �b�

� � �Á ~ � Á � � � Á � � � � ~�² ³�² ³ �� � � �

��

��

� �

� for some . Thus, if , then letting , there exists� �

� � � such that

� � � � � �

� � �

� � �

²� !³ b ²� !³ ~ ²��² ³!³ b ²��² ³!³ ²�À��³

~ ² b ³ ²�!³

~ ²�² b ³�!³Á

� � � �

� �

� �

so that is scaling stable.@Conversely, assume is a scaling stable r.v. Let (7.27) be its log ch.f. and @ ~ ¸@ ¹$

be a singleton space with measure one. For each , let be independent copies� h� ¸@ ¹�� �

of in which are assigned measure , and let . Then @ ² ³ ~ : ~ @ ² ³$ � � � � �� � ��

� � � �� � �

�

has ch.f.

� � �

�

� � �

� �

�� �

� �

²!³ ~ c � ! b �� ! ²�À��³!

O!O

~ c � O!O b �� O!O Á!

O!O

~ c � O!O b �� O!O Á!

O!O

� e e e e: ;: ;��

� �� �

�

� � �

� �

� �� �

so that the distribution of is independent of . Letting also denote the identity: @� �function on , this shows that in distribution. $ � �

$ �@ ²� ³ ~ : ~ @ Êlim�¦B

We now show that the scaling function must be very restricted for a (non-trivial)�Lebesgue integral to exist. To properly motivate this, we make some observations aboutso-called semi-stable stochastic processes [La]. A stochastic process on is ?! l semi-stable if for every with , and denoting� � �Á ? � �²�³? b �²�³ �Á � � ��! ! lisomorphism. We assume is continuous, i.e., that?!

lim�¦�

!b� !F �²O? c? O � ³ ~ �Á

and that is proper, i.e. non-degenerate for all . We then have? !!

Theorem 7.12 [La] ¢ If is semi-stable, proper, and continuous in the above sense,?!

and if has independent increments, then?!

�²�³ ~ � Á�

for some .� � �

92

With the proper interpretation, this may be viewed as a special case of the nexttheorem. The identification depends on the fact that any semi-stable, stationary 0-meanstochastic process with independent increments can be written@!

@ ~ ?²!³ ²�!³Á!cB

!� �

where is a measurable r.v.-valued function and .? ¢ ¦� l lb b

Theorem 7.13: If exists and is non-zero, then there exists such@ ~ ? ²� ³ � �3$ �

� � �

that for some . Specifically, is given by (7.27), where is the log� �

�

² ³��

­ � �� ¦ �

� � � �

ch.f. of .@

Proof: Note that convergence of implies . Let . Since@ ² ³ � ~ ²?³� � ­ : H� ¦ �

� ess

@ ~ ²� ³ ² ~ ? ³:��� � � �i i c�exists , it does also (and is non-zero) on some bounded,

finite sub-domain of Thus assume without loss that is -bounded and -finite.: : � �� �iÀ �

Let be constructed as in (7.29). Let each contributing to be subdivided into : 7 : �� �� �

equal pieces, each containing (again, arguments become technical if formal?��

subdivision of is not allowed). This subdivision gives?��

: : ;; : ;� � 6 77 ¤7 7 ¤7

�� ��

c�i� i�

c�

��

c� �¦B�� ��� �

i i

c�

? ~ ? ²� ³ ¬ @ Á ²�À��³�

� ²� ³� �

�

�

so that, letting � ¦ BÁ

@ ~@

�²�³

i�

in distribution, where By (7.27), therefore, . Thus,�²�³ ~ À �²�³ ~ �lim � �

� �

²� ³² ³

��

�6 7�� � �� � � �c �� for some .

We now assume is not asymptotic to , for a contradiction. Assume without� � �² ³ ���

loss that for some sequence , for . Let .� � � � � � �� � ��c ¦ �Á ² ³ � ²� b ³ � � ~ �

�

��

Then by the proof of Theorem 7.11,

: ~ ² ³ ? ¬ @ À ²�À��³ � �

7 ¤7�¦B

� �

�� �i

� � �We assume without loss that the numbers (see proof of Theorem 7.11) have the��property that is integral.���

Let be defined by , where denotes greatest integer. By readjustment � ~ ´� µ ´ h µ� � �

of in the proof of Theorem 7.11, we assume without loss that and lie in a¸� ¹ � � �

single interval . Furthermore, we can choose such that appears in´� Á � ³ 7 ¸? ¢ ?� �b� � � �

93

: ¹ � ¸? ¢ ? ¹ � � ? ~ ¸? ¢ ? � � � �b� � � �� � � appears in S for . If appears in˜

: � � : ¹ ? � �b� � for some , but not in (we treat all distinct elements of as˜�

independent), then

: ~ ? ²� ³ ¬ �À ²�À�³˜�

?�?

c

�¦B�̃

�

��

Since the argument for this is similar to one for in the proof of the last theorem, we:�i

omit it.It follows from (7.34) and (7.35) that

�²� ³ ? ¬ @ Àc

7 ¤7

��¦B

�

�� �i

��

This provides the contradiction, since

�²� ³ ? ¬ @ Á ²�À�³c

7 ¤7

��¦B

�

�� �i

��

and by assumption fails to converge to 1 as �

�

²� ³²� ³

c �

c � � ¦ BÀ Ê

It follows from Theorem 7.13 and Corollary 7.3.1 that the set of admissible functions� (i.e., measurable functions yielding non-trivial integrals) fall into equivalence classeswith if for some constant . Furthermore, each class has exactly� �� �

² ³² ³

� � �� �

� ��

�­

� ¦ �

one homogeneous representative . Thus the classes of are indexed by the� � � �² ³ ~��

positive reals.

94

Chapter 8

Integrability Criteria and Some Applications

In the first half of this chapter , we introduce an important integrability criterion forrandom variable-valued functions (Theorem 1). We then compare the Lebesgue with theRiemann integral, and finally give applications to the calculation of asymptotic jointdistributions of commuting observables.

A central result shows the non-linear Lebesgue integral of Chapter 7 to becontinuously imbeddable into a standard Lebesgue integral over the space of0 4i i

measures on the Borel sets of , the space of probability distributions. Let be the space: 9of logarithms of ch.f.'s (log ch.f.'s) of infinitely divisible distributions. Let be1 ¢ ¦: 9the partially defined function , where . Let be1 ~ ²� c �³� ~� � � � �lim

�� � � � �¦�

�!% � %² ³

4 5a measure on the Borel sets of , and denote the integration operator: 90 ¢ 4 ¦i i

0 ~ 1 � ² ³Ài � � � ��Finally, let denote the operation taking a ch.f. to its probability distribution, and < , ¢9 : � <¦ , ~ ²� ³À be given by �

Theorem 8.5 states that if denotes the Lebesgue integral0 ¢ 4 ¦�i :

0 ~ ²� ² ³³Á�:

� � � � �3�

then the diagram

95

commutes. Specifically, the Lebesgue integral of a r.v.-valued function is? ¢ ¦$ :

3� �: ;6 7$ $ �

� � �? ²� ³ ~ ²� ³ c � � ² ³ À�

� � < ; � ��

exp lim¦�

�?² ³! ² ³

§8.1 The Criterion

The metric (equation (7.21)) on the space of probability laws will be used� :�

throughout this chapter. Let be measurable from a space and? ¢ ¦ � ² Á Á ³$ : : $ 7 ��

� � � � � � � � �i c�� � � � � � �

� �

i~ ? À Á ~ i ià  ~ b b ÀÀÀFor measures let as usual, let� �We say is - if it is -separable. Note that -Lebesgue integrability is: � � �� separable �

senseless if or some subset containing the essential part of the range is not -: ��

separable, since required partitions cannot be countable. A function is7 ? ¢ ¦� $ :bounded if its range has finite -diameter. We now show that, in complete analogy to��the real-valued case, if is finite and is bounded and measurable, then it is� $ :? ¢ ¦ �

Lebesgue integrable. We require

Lemma 8.1.1: Let , be two countable families of finite Borel measures on²�³ � ��� ��

l. Then

� � � � � � � � �3 3 3

� � �

� �� � �� � �� �� �� ���

: ; : ;� � �� Á � � � ² Á ³Á ² Á ³ À ²�À�³max sup

(b) Similarly, if is a measure space and are d.f.'s, then² Á Á ³ - ²%³Á . ²%³: 7 �i� �

� � � � �3 i i: ;� �: :

� �- ²%³� ² ³Á . ²%³� ² ³

� ²- Á. ³� ² ³Á ²- Á. ³ Àmax sup: ;�:

� � � ��

� � � �3 i 3

Proof of a :² ³ Set , and let be the right side of (8.1). Letting be the� � � � �� �� �� ��3~ ² Á ³ -

corresponding distribution functions,

� � ���

²� - ²% c ³³ c � � ²- ²% c ³ c ³ � � - ²%³

� � ²- ²% b ³ b ³

� ²� - ²% b ³³ b ÀÊ

� �� � �� � � � ��

� �� � �

� ��

� � � �

� �

� �

96

For a function from a measure space to a topological space recall that is in? �$ : � :the essential range if for every neighborhood of .H � �ess²?³ ? ²5³ � � 5c�

Recall also

� � �²%³ ~ Á ²%³ ~ Á ~ À ²�À�³% %

� b % � b %

%Â O%O � �

 O%O � �

�

� �i

�%

H

Theorem 8.1: Let be finite, and , be bounded and -Borel² Á Á ³ ? ¢ ¦$ 8 � $ : ��measurable. Then exists if and only if@ ~ ?² ³ ²� ² ³³

3$

� � � �

.²  %³ � � ²&³Á � � ²&³ ²�À�³& &

� b & � b &� � � � - w lim lim

� �� � �

¦� ¦�cB cB

% B�

� �� �exist for every in the essential range of , where . Furthermore, the� � �? ~� � � �

� %² ³6 7

pa r�

� � � � � �~ ² ³Á .²%³ ~ .²? %³� ² ³� �?

is the Lévy-Khinchin transform of .@

Proof: Assume and exist for supp where . Let � � � � � � �² ³ .² ³ � ~ ? ¸7 ¹i i c� B~�� �

be an infinitesimal sequence of partitions of For , we first: H �� � �~ ²?³À � 7ess � �

verify that the distributions form an infinitesimal sequence. To this� %² ³ ² ³�� � � ��

� �� �� 6 7

end, we need to show uniformly in . Suppose this is� %² ³ ² ³

¦B�� � � �

�� � � H6 7 ¬ � ²?³ess

false. Then there exists such that for every , there is a , such� � H� � 5 � � � ²?³ess

that ( . This contradicts the boundedness (i.e., finite� � �cBÁ c5³ b ²5ÁB³ �

diameter) of since then can be made arbitrarily largeH � �ess²?³Á ²&³�sup�

� � �� %

cB

B

² ³ 6 7

for Thus the sequence is infinitesimal.� H� ²?³Àess

According to Corollary 7.3.1, it now suffices to show

� � : ;�

� �cB

B�

�� b � ²�À��³

& b �� �

�

�

� � ­ �� �¦B

�� : ;� cB

%

��

� ¦B� �

�� ¬ .²%³Á ²�À��³

& b ��

�

� �

97

where , and� � �� �� �~ ² ³

� ~ ²&³� À ²�À³&

� ��

� �cB

Bi

�� : ;� �

�

First consider (8.4b). Note that

� �� �: ; : ;� �cB cB

% %b�

� ��

� �� � � �

� � � �²&³ � ~ ²&³�

& b � &

² ³ ² ³� �

�

� �

��

c � ²& b � ³� ²�À³&

² ³� ��

�� �

cB

%b�Z i

��

� : ;��

� �� �

where . In addition, is bounded uniformly in and (for O� O � O� O � �� � ���i

� ����

� �

sufficiently large) for all non-vanishing , since the support of is bounded and� ���i

replacing in the metric by gives an equivalent one (see the remark after� � ��i

Proposition 7.4.). The second term in the sum on the right of (8.6) converges to 0

uniformly in and since are infinitesimal, so that% � F 6 7G�� � ��%

² ³���

� � � � � ­� � � �

3

� �cB cB

% %

� ��

� �: : ; : ;;� �� �� Á � �À ²�À�³

& b � &

² ³ ² ³� �

�

� ��¦B

Let be the d.f. of . Then. ²  %³ ~ ²%³� �� �� � �� � � � ��

cB

% &² ³

6 7� � � � � �

� �3 i

� cB

%

��

: : ;;� ��:

���

.²  %³� ² ³Á �&

² ³

� .²  %³� ² ³Á .²  %³� ² ³ ²�À�³� � � � � � �3 i i

� �7 7�: ;� �� �

� �� �

�

b .²  %³Á . ²  %³ À� � � � �3

� �

� � � �: ;� �� � � � ���

To show (8.8) vanishes as , we note the first term on the right vanishes by Lemma� ¦ B8.1.1, if

� � �3�².²  %³Á .² Á %³³ �� ­

�¦B

for , uniformly in and ; the latter follows from the metric infinitesimality of� �� 7 ���

the partition , and the equivalence of and for fixed .7 � �� � �� � �3Á

98

The second term on the right requires some care, however. For , , let� � � �, ~ ¸ � ¢ ². ² ³Á .² ³³ � � ¹ %�� �� : � � � � � �� �

3�

for ; note we have suppressed . Let, ~ ¸ � ¢ ² Á , ³ � ¹ , � ,�� � � �� �� �� � � Â� : � � � � � be the -neighborhood of . Let and� � � � � � ��

Z Z� � , ² Á ³ � � Á. If , and we have�� �

� � � � � � � � � � � �

� � �

� �

3 3 Z 3 Z Z 3 Z². ² ³Á .² ³³ � ². ² ³Á. ² ³³ b ². ² ³Á.² ³³ b ².² ³Á.² ³³� � b b �~ b � Â

²�À ³� � � �� � � �

we have used the definition (7.21) of in terms of Thus� � � � � �� � �² Á ³ ². ² ³Á . ² ³³À� � � �3

, � , À ²�À��³�� � � � � ² b� ³

Given , let , diam . By the Lemma (� h � � � �� ~ Á ~ � ~ � ²7 ³ ��� � � �� � �sup sup

denotes complement)

� � � � �3

� �

� � � �: ;� �� � � � �.² ³Á . ² ³��

� ²7 q , ³.² ³Á ²7 q , ³. ² ³ ²�À��³� � � � �3 i i

� �

� � � �: ;� �� �� � � �� � ���

b ²7 q , ³.² ³Á ²7 q , ³. ² ³ À� � � � �3 i i

� �

� � � �: ;� �� �� � � �� � �˜ ˜

��

If , then by (8.10), since diam . Hence7 q , £ 7 � , ²7 ³ �� �� � �� � ��

� � �² b� ³ ��

� � � � � �� � � �� � � �� � �² b� ³ ¸�¢7 q, £ ¹3� , ².² ³Á. ² ³³ � b � �, and , so thatsup

� �� �� �­�¦B

the first term on the right of (8.11) vanishes as by Lemma 8.1.1. The second� ¦ BÁterm, on the other hand, is bounded by . For ˜� � � � :i

�� � � � �², ³ . ² ÁB³ Á � Á�� � � : �sup� � �

. ² ÁB³ c . ² ÁB³ � ² ³ � BÁ� �� � :� � �diam

so that . Since , this term vanishes as so˜sup� � :

� ��

� �

�

�� �

i. ² ÁB³ � B ², ³ � ¦ BÁ� � �­�¦B

that (8.8) vanishes as . Together with (8.7) this proves (8.4b) (recall� ¦ B $.²?Á %³� ² ³ ~ .² Á %³� ² ³³� � � � �

Hess²?³ .i

We now prove (8.4a). By the mean value theorem,

� � : ;�

� �cB

B�

�� b �

% b �� �

�

�

� ��

~ � b � ²� c ²% b � ³³� Á% %� �� �: ; : : ;;

� �cB cB

B B

� � �� �

Z i�� � � �

� �� � �

� ��

99

for some By the arguments for (8.6), the last term vanishes as , andO� O � O� OÀ ¦ B� �i �

we are left with proving

�� : ;� cB

B

��

� � ��

� ¦ Â%

��

which also follows along the same lines as the first part of the proof.We now prove the inverse of the above, namely, that (8.4) is necessary for

convergence. Suppose first that for some supp fails to exist. Define � � � �� Á .² ³ � �i

by

�

� :� � �

i� ¦� Á �

3

² ³~ ². ² ³Á. ² ³³À ²�À��³lim sup

� � � �� �

� �

� �

Let . We show that fails to exist, and5 ~ ¸ � ¢ ² Á ³ � ¹ ²� ³� : � � � �� �Z Z i� � 5�

�3

hence (see Proposition 7.6) the integral over fails to exist. To this end, we may:�

assume that . Let be an infinitesimal sequence of partitions of . In the5 ~ 7: :� ��

following, we may assume the partitions are even without loss, by the² ~ ³� �� �� �

arguments of Theorem 7.11 (i.e., the total measure of those elements for which7���i

�²7 ³� does not equal the common value can be made arbitrarily small). Then (letting� �� � be the common value of )�

� � � � : � � � � � �3 i 3

� � �

� � � �: ; : ;� � �� � � � � � � � �. ² ³Á ² ³. ² ³ ~ . ² ³Á . ² ³� � � ��

� ². ² ³Á. ² ³³Á ². ² ³Á . ² ³³ ²�À��³

� ² ³Á ~ Á� � �

max sup: ;�: ;

�

3 3� �

�

i�

� � � � � � �

� � �� :

� � � � � � �� � � �

max

assuming (without loss) that � :i�² ³ � �À

Again without loss of generality, we may assume by (8.12) that the sequence is��such that

� : � � � �i 3�

¦B Á �² ³ ². ² ³Á . ² ³³ ~ À ²�À��³lim sup

� � � �� �

� �� �� �

Hence by (8.13)

lim sup� � � �

� � � � � �¦B Á �

3

� �

� � � �� �

� � � � � �� �� � � � �

�: ;� �. ² ³Á . ² ³ � Á ²�À�³�� �

so that fails to converge, and by (8.6), so does��

� �� �� � �. ² ³��

100

��

� � �� � �� � � �. ² ²% b � ³³ ¦ BÀ��

. (The last term in (8.6) still vanishes as ) Thus

3 �

:�� �²� ³i fails to exist by Corollary 7.3.1.

Now assume fails to exist. If the weak limit of exists, then� � �� ��

� � �cB cB cB

B B Bi� � � � � �²%³� ²%³ ~ ²%³� ²%³ b �²%³ ²%³� ²%³Á ²�À�³� � �

where so that the left side of (8.16) also fails to converge as . From� � * ² ³Á ¦ �) l �here on the argument using Corollary 7.3.1 is the same as above, and again

3 �� �²� ³i

fails to converge. This completes the proof. Ê

The above conditions can be simplified to a large extent.

Definition 8.2: For denotes those functions defined for small and� l 7 � � �Z b� Á ² ³�Z

positive satisfying . Let . Define .� �

� � �² ³

�f

�Z Z ��

Z­ l 7 7 l l� ¦ �

* � ~ � � ¸�¹�Theorem 8.3: Let and satisfy the hypotheses of Theorem 8.1. Then?Á Á� �@ ~

3�? ²� ³� � exists if and only if

²�³ � �� 7 ��Z for some Z ��

²��³ ? for in the essential range of , converges weakly to a measure on � � l�f

as � ¦ B

²���³ � Á ²�³ ~ ² ³ � B ²�³ � � �Á ² ³ ~ �for supp if then if ; and� � � L � � ; �i Z Z� �� �

²�³ ~ �Á % � if exists.� �Z¦B c4

4lim� Note that the weak convergence condition above means converges to (possibly��

infinite) measures on and , individually.l lb c

Proof: We first show the above imply the necessary and sufficient conditions of the

previous theorem. Let w- w- . Simple scaling arguments8 � �� � �lim lim� � � �¦�� %6 7�Z

(replace by and let ) show that is homogeneous, and that% �% ¦ � 8� �

�²8 ³ ~ � O%O b � O%O �% ²% � ³Á ²�À��³%

O%O� l: ;� �

c c� c c� f� �

where . For , let . Then since� � �~ % � � - ²%³ ~ ´%ÁB³Â - ² c %³ ~ ² cBÁ c %³� b c�Z

101

� % � b �- % Á� � �

­b c� �: ;�

�Z

� ¦ �

it follows that

% - ²%³ Á� b �� b � �

­�% ¦ B

and similarly

²%³ - ² c %³ À ²�À��³� c �� c � �

­�% ¦ B

If , then for supp � � �Z i��� � Á

� �c� c4

� 4� c� �% � ~ 4 % � ²%³Á ²�À� ³� ��

�

where (8.19) is easily shown to converge using the asymptotics of and ,4 ~ Â - -� b c�

after integration by parts. Hence converges weakly on and thus % � ´ c �Á �µ ²%³�� � � �� �

does on , so that the first weak limit in (8.3) exists. To examine the same limit whenl� L �Z �

�~ ² ³, notice that (8.19) now converges by the finiteness of .We prove existence of the second limit in (8.3). If , the limit follows from the�Z � �

convergence of

� �c� c4

� 4c�% � ~ 4 %� ²%³Á ²�À��³� ���

via integration by parts as before. Then ,��

Z� � ��

4 %� ~ c4 %� b %� Á� �c� c�

c4 4 cB

4 B c4� � �B C� � �b

c

and the latter again converges via integration by parts. This completes the proof ofsufficiency.

We now assume (8.3) to hold, and prove - ( is clearly necessary by²��³ ²���³ ²�³Theorem 7.11). First, follows immediately from convergence of , since weak²��³ .²  %³�convergence of on implies the same for . To prove , first let .� � l � �� � ²���³ ~� �

f Z ��

Since converges weakly by hypothesis,� �� �

lim lim�

�¦� c� c4

� 4� �

4¦B� �% � ~ % �� �

converges, proving (a). Similarly, if , then²���³ � � ���

Z�

102

lim lim�

��

¦� c� c4

� 4

4¦B

c�� �% � ~ 4 %� À ²�À��³� �

Since this converges (by 8.3) and it follows that . To show� �c � � �Á % � ~ �lim4¦B c4

4that indeed has a first moment, we show (since the argument on is� � l

�

B c% � � B

the same). Let be a monotone function, , defined by . By (8.19). .²�³ ~ � �. ~ %��and (8.3), we have convergence as of4 ¦ B

4 % � ~ 4 ².²4³ c (²4³³Á� �c� � c�

�

4� �

where

(²4³ ~ .²%³�%À ²�À��³�

4��

4

Hence

.²4³ c (²4³ � �²4³ ²4 � �³Á

where is a smooth positive function, with �²4³ �²�³ ~ �Á � ²�³ � * ÁZ�

�²4³ ~ * 4 4 � � .²%³ � %��c� for ; note that . There exists a monotone function

. Á . ²�³ ~ �i i , satisfying

. ²4³ c ( ²4³ ~ �²4³Ái i

where and are related by (8.22). The equation for ,. ( 4 � �i i

. ²4³ c ( ²4³ ~ * 4 ²�À��³i i �c�

�

is solved by differentiating once, giving

. ²4³ ~ * b * 4 ²4 � �³Á� c

� ci c

� �: ;�� �

so that is bounded. On the other hand,.i

². c .³ c ²( c (³ � � ²�À��³i i

on this implies for , so is bounded, provinglb i i ². c .³²4³ � ². c.³²�³ 4 � � .that has a finite first moment on . The proof is of course the same on . Thus� l lb c

necessity of (b) is proved; necessity of (c) for convergence of the second²���³ ²���³integral in (8.3) is clear, and this completes the proof. Ê

We now consider the general (non-finite) Lebesgue integral on a measure space² Á Á ³$ 8 � . Recall (8.2).

103

Lemma 8.4.1: Let be a sequence of independent r.v.'s, and . The sum? � ~ ²? ³� � �i;�

� � ��~�

B

� � � �? O� O ²? c � ³ converges order-independently if and only if and ;�

converge.

Proof: Assume the numerical series converge absolutely, and write, using the mean valuetheorem,

� � �� � �

� � � � �Z i

�;� ; � ; �²? b � ³ ~ ²? ³ b � ²? b � ³Á ²�À�³

where with probability one. It is easy to see the are infinitesimal, and inO� O � O� O ?�i

� �

particular that , so by (8.25) converges. From this it; � ­ ; �Z� � �

�¦B²? b � ³ � ²? ³�

follows (with convergence of ) that converges as well, and sufficiency� �O� O ²? ³� �;�

follows by Lemma 7.3.1. Conversely, using Lemma 7.3.1, if and � �; � ;�²? ³ ²? ³� �i

converge absolutely, so does , and converges by (8.25). � �;� ; �²? ³ ²? c � ³ Ê� � �

Recall that, for a measure on and , the measures and are� l � l l � �¢ ¦ .b b�

� � � � �� � �

� ��

~ Á �². ³ ~ � Á ²�À�³� %

² ³: ; w- lim¦�

and

� � � � � �� � �� �

�~ ²%³� ²%³Á ~ ²%³� ²%³ ²�À��³lim lim¦� ¦�cB cB

B Bi i� �

(see (8.2)), when the limits exist. If is an r.v., and are defined analogously by? .? �?the distribution of . If is a measure, denotes total measure.� � �? O O

Theorem 8.4: Given : , and , a random variable-valued function� l l $ :b b¦ ? ¢ ¦on the measure space , the integral exists if and only if $ � � � �@ � ?² ³ ²� ² ³³ ²�³3 $. ? ²�³ O.?² ³O� ² ³� � � � � � and exist for every in the essential range of , and and� $ e e$ $ $�� � � � � �?² ³ ?� ² ³ ² � Á .?� ³converge. Furthermore, the pair is the Lévy-

Khinchin transform of .@

Note that we make no restriction on , i.e., may be measurable map into the: ? anyspace of probability distributions. This theorem, together with Proposition 7.6, showsthat an integrable induces a countably additive measure-valued measure defined by? CC � �²)³ ~ ? ²� ³

).

Proof: Assume (1) and (2) hold. Let be the measure induced on by . Since� :i ?

O O � � O O b � O. O � Á � � �Á �� � � � �� � : �i i i

� � � � ?² ³ for some converges as well. Let e e

104

¸7 ¹ ~ ²?³� ��0 � be an at most countable partition of into sets of finite measure and: Hess

diameter. Let

: � : � : � : ��� � �� �i i~ ¸ � ¢ � �¹Á ~ ¸ �  � �¹À� �

Assume without loss that respects the partition of . Let be an at7 Á ¸7 ¹: : :�� �� � �� �

most countable partition of whose elements have finite measure and diameter, with7�

� � � � � � � ��� �� �� �� �� �� �� ��i i

cB

B %� 7 Á ~ ²7 ³Á ~ ² ³ � ~ ²%³�. Let . By the proof of 4 5���

Theorem 8.1, can be made to differ from by arbitrarily little; hence� �

�� 7i iO� O O O� ² ³

�� � ��

the sub-partitions can be chosen so that converges. Given , for a7 O� O � � 0�� ����0Á�

�sufficiently small sub-partition of (still denoted by is unchanged),¸7 ¹ ¸7 ¹Â 7�� � �� �� �

i

�� �� 7i� � �� � � can be made arbitrarily close to . Hence by Theorem 1,

3 �²� ² ³³ �À

� � �: ;� cB 7

B

����

��

i� � � ��

� c O. O� ²�À��³% b �

�

can be made arbitrarily small. Therefore by condition (2), there exists a sub-partition

¸7 ¹ � � 7�� �Á� �� �� ���Á� �Á�

cB

B %b� such that and converge absolutely under , as well� � 6 7� � ��

���

as any finer partition. Thus, by the Lemma, converges order� �6 7� �

i i

�� ��� �

independently. Since the inner sum approximates arbitrarily well, the sum3 �

7

i�� �²� ³

� �

i

7i

3 3��� � � � � �²� ³ ~ ?² ³ ²� ² ³³

$ is also order independent. Since any two

partitions have a minimal common refinement, this sum is also independent of ¸7 ¹À�

Conversely, if exists, we claim . For if the latter is3 $ $? ²� ³ O.?O� � B� � �

false, let be an at most countable partition of into sets of finite measure and¸7 ¹� ��0 $

diameter. The sum diverges; and therefore must also fail� � � �

7� �O.?O� ? ²� ³� � �

�

to converge by the linearity and continuity of the Lévy-Khinchin transform (Remarksafter Theorem 7.1). Thus the integral fails to exist. A similar contraction obtains if$ �O O� ~ B� �i .

Finally, if exists, then , where@ ~ ? ²� ³ @ ~ @3 ��

�$

� �

@ ~ ? ²� ³�7

3

�

� � �

and is a sequence of subsets of with finite measure and diameter. Thus, the Lévy-7� $Khinchin transform of is the sum of those of . This together with the final assertion@ @�

in Theorem 8.1 completes the proof. Ê

105

We remark that conditions (1) can naturally be replaced by conditions - of²�³ ²���³Theorem 8.3.

The following lemma is proved in Loève [Lo, §23].

Lemma 8.5.1: Let , and be finite Borel measures on . If� l * l� ��

�! b � c � c �� *� �cB

B �!% �!%�b% %

²�b% ³ 4 5� �

�

converges to a function continuous at theorigin, then converges and converges weakly.¸ ¹ ¸ ¹� *� � � �

Theorem 8.5: The integral exists if and only if@ ~ ? ²� ³3$

� �

²�³ ²!³ ~ ² ² ² ³!³ c �³ ! ~ � � ²?³Á� ) � � � H� ��

�lim¦�

� exists and is continuous at for ess

where is the characteristic function of ) ��

²��³ the integral

� � � �²!³ ~ ²!³� ² ³ ²�À� ³�$

�?² ³

converges absolutely.

In this case has characteristic function @ ��.

Proof: If exists, then according to Theorem 8.4, so do and for @ . � ²?³À� � � H� ess

Furthermore, if � H� ²?³Áess

lim�

� �¦�

�!% �%!� �

�� � : ;²� c �³� ²%³ ~ � ! b � c � c �². ³²%³�%! � b %

� b % %� � �

is clearly continuous at 0, proving . According to (7.1), has the log ch.f.²�³ @

� ��cB

B�%!

� �

�

²!³ ~ � ! b � c � c �. ²�À��³�%! � b %

� b % %� : ;: ;

where, by Theorem 8.4,

� � � � �~ � Á . ~ . � Á� �: :

�i i

and is the induced measure on . Since the integrand in (8.30) is bounded, we can� :i

interchange integrals, so

106

� � � � �

� � � �

�

�cB

B�%! i

� �

�

¦� cB cB

B B

� ��%! i

¦�

²!³ ~ � ! b � c � c �². ³²%³ � ² ³�%! � b %

� b % %

~ �! � b � c � c � � ² ³% �%!

� b % � b %

~�

� �B : ;: ; C� � �B : ; C� B

:�

: �� �

: �

lim

lim � CcB

B�% ! i²� c �³� ²%³ � ² ³ ~ ²!³À

²�À��³

� � � � �

It is clear that the integrals converge absolutely.Conversely, assume that and hold. If ²�³ ²��³ � ²?³Á� Hess

lim lim� �

� � �¦� ¦�

�%! �!%� �

�� � � : ;: ;²� c �³� ~ �! � b � c � c ²%³��!% � b %

� b % %� � � � �

is continuous at 0, so that by the Lemma, and w- exist.� � � � � �� � �~ � �. ~ �lim limLet be a countable partition of into sets of finite measure, on each of which ¸7 ¹ ?� ��0 $is essentially bounded. This can be done by appropriately partitioning . LetHess²?³$ � �� ��~�

�~ 7 ? ²� ³ �� . By Theorem 8.4, exists for all , and by part one of this$�

proof, if is its log ch.f., then��

� � � �� ?² ³~ � ² ³À ²�À��³�$

�

�

Clearly, converges weakly as to an r.v. with ch.f. . Thus is$�? ²� ³ � ¦ B @ ²!³ @� � �

independent of the choice of , and is the desired integral. ¸7 ¹ Ê� ��0

§8.2 The Riemann Integral

Let be a metric space with -finite Borel measure , and be an² Á ³ ? ¢ ¦$ � � � $ :r.v.-valued function. The Riemann integral of is defined with respect to partitions of ? $rather than . We might again try to duplicate the elegance of the Lebesgue theory in the:Riemann integral, but we approach the latter with a view to utility, namely, to physicallymotivated applications. With this intent, we define the Riemann integral using globalpartitions of , rather than patching integrals over sets of finite measure whose range$under is bounded. The former definition will dovetail with that of the -integral.? 9i

Definition 8:6: Let be an infinitesimal sequence of partitions of . Select a¸7 ¹� �~�B $

function : . Let , and assume� l l �b b� �¦ � 7� �

��

� �¦B

?² ³ ² ³ ¬ @ Á� � �� ��

where , and the are independent. If is independent of { },� � �� � �� $� � �~ ²7 ³ ¸?² ³¹ @ 7

107

then is the - of (with respect to the non-linear measure ),@ ² ³ ? ²� ³� � �Riemann integral

@ ~ ?² ³ ²� ² ³³ À ²�À��³9�

$

� � � �

Theorem 8.7: If is -Riemann integrable then it is -Lebesgue integrable, and the?² ³� � �integrals coincide.

Proof: Recall (7.21). We begin by assuming is finite and is -bounded. Let be� �? 7� �

an infinitesimal net of partitions of . Recall the Lebesgue integral may be equivalently$defined by allowing partitions of to subdivide individual elements, even ones of:measure 0. Under this more general (but equivalent) definition, there exists a partitionsequence of such that . As before, we may use the arguments of7 ?²7 ³ ~ 7� ��

i i� �:

Theorem 7.11 to assume (without subsequent loss) that the partitions are even7 ²7 ³� �i

with respect to , i.e., ; again the measure of those� � � � �² ~ ? ³ ²7 ³ ~ ²7 ³i c�� �� �

elements with odd (unequal) measure can be made arbitrarily small by taking sufficientlysmall sub-partitions. By taking sub-partitions, we may also assume that vanish in7��

i

diameter uniformly as .� ¦ B

We proceed by contraposition, assuming the Lebesgue integral fails to exist. In thiscase it suffices to show the Riemann integral fails to exist for any

$�? ²� ³ �� � $ $�

with positive measure, by arguments used in proving Proposition 7.6. By our assumptionand Theorem 8.1, either or fails to exist for some . Assume does not. � .� � � H �� ess

exist. At this point, using the above net of partitions, the argument becomes exactly7�i

analogous to that after (8.12), so we omit the details. The same argument works if ��fails to exist, yielding the result when is finite and is -bounded. The fact that the� �? �

two integrals coincide in this case follows from correspondence of the Riemann sumsover the partitions and .7 7� �

i

If is infinite or is unbounded, and exists, then the integral also� $ � �² ³ ? ? ²� ³9$

exists over any subset with positive measure. Since the essential range of is -$ $ �� � ?finite, it is separable. Let be an at most countable partition of , with and¸ ¹ ² ³: : � :� �

i

diam . If , then exists, and we must show that ² ³ � B ~ ? ² ³ ? ²� ³: $ : � �� � �c�

�3 �

�$

3 9�

$? ²� ³ ?²� ³� � � converges order-independently to . This follows from the fact

$

that can be approximated arbitrarily well by Riemann sums over partitions3 �

$? ²� ³� �

of whose total sum over approximates $ � �� � ? ²� ³À Ê9$

Recall that for ,� :�

. ²  %³ ~ ²&³�  ² ³ ~ ²&³� À� & � &

² ³ ² ³� �� � � � � � �

� � � � � �� �: ; : ;cB cB

% B

We now prove

108

Theorem 8.8: Suppose

²�³ ? ¢ ¦ is Lebesgue integrable and -continuous,$ : ��

²��³ ² ³ � . ²?² ³ÁB³ b O ²?² ³³O 3 ² ² ³³ the function g is in � � � �� �

� � � �

Á � ��

² Á ³�

�� � � � � �sup�

�

� �

for some , .� � � �

Then is Riemann-integrable.?

Proof: We prove this for partitions which do not subdivide points (the proof thereforedoes not directly work for measure spaces with atoms). The general case (which allowspartition elements which overlap on atoms) follows with small modifications.

Let be an infinitesimal net of partitions of , and . Let7 � 7� � �$ � � �

? ~ ?² ³Á ~ ²7 ³Á ~ ² ³� � � � � �� � � � � �� � � � � � . We first verify that the double sequence¸ ? ¹�� �� � is infinitesimal. Suppose it is not. Taking a subsequence if necessary, we mayassume without loss of generality that

� �� � �� � �� � ��. ²? ÁB³ �

�

for some , and some choice of for each . Therefore if ,� � � �� � � � � �Á�

� ² ³ � ² ² Á ³ � ³ ²�À��³� � ��

Á ��

�� � � � �

�

��

�

for sufficiently large that . Furthermore, again taking subsequences if� � ���� �necessary, we assume that . Then if (the -ball about� � � �� � � ��� �

�� ² c�³�� ��

� ) ~ ) ² ³c�

����),

� � � �: ;$

� � � �� � ² ³ � b b bà � �Á Á) ) �) ) �²) r) ³

� � �� � � � � �

� ²) ³ b c ²) ³ b c ²) ³ bÃ

� ²) ³ b ²) ³ b ²) ³ bÃ� �

� �

~ BÁ

� � � � �

� � � � �� � �

� � �

� � �� � �

� � � � �

�� �� �� �� ��� � �

� � �

�� �� ��� � �

� � � � �

� � �

: ; : ;

since for sufficiently large; this gives the desired contradiction. Thus,� � �²) ³ �� ���

¸ ? ¹�� �� � is infinitesimal. To prove the theorem, according to Corollary 7.3.1, it suffices to prove the

convergence

109

�� : ;� cB

%

��

� ¦B� �

�� ¬ .²%³Â ²�À��³

& b ��

�

� �

��

�¦B

Z� Á ²�À��³��­ �

for some a multiple of a d.f., and some , both independent of .. ¢ ¦ � ¸7 ¹l l � lZ�

Here, is the distribution of , and�� �� �?

� ~ � Á ²�À�³%

� ��

� �cB

Bi

�� : ;� �

�

with given by (7.9). We have�i

�� �: ;� cB

%

��

�� � � � �

�� ~ / ²  %³� ² ³Á

& b �� �

�

� $

where

/ ²  %³ ~ � ² � 7 ³À� & b �

� � �� �

�� � � �

� �� �cB

%

� ��� : ;

Since is -continuous, if supp , then so that by (1) of? � ² ³ ?² ³ � ²?³Á� � � � H� ess

Theorem 8.4,

/ ²  %³ ¬ /²  %³Á ²�À��³��

� �¦B

where is a multiple of a d.f. in for each . Furthermore,/²  %³ %� �/ ²  %³ � / ² ÂB³� �� �

~ � b ²%³� ² � 7 ³Á ²�À��³� & &�

� � � �� � � � �

� � � �� � �

�

� � � �cB cB

B B

� � �� Zi� �: ; : ;

where is determined by the mean value theorem. The first term on the rightO� O � O� O� ��i

�

is clearly dominated by for sufficiently large. Since the double� ² ³ � 3 ² ³� �Á�� � �

sequence is infinitesimal and , the second term on the right is� �� ��% Z4 5��

²�³ ~ �

eventually dominated by

/ ² ÂB³ ~ ~ � � 2 � ² ³ ² � 7 ³Á ²�À� ³� � %

��

� � �� � � �

²�³ i�

� � �cB

B

� Á �� � � � �� � �

� : ;for sufficiently large, with independent of Thus, for sufficiently large,� � �2 À

110

/ ²  "³ � ²� b2³� ² ³ � 3 À ²�À��³� � �� �Á�

By (8.37), if �²%³ � * ² ³Á�B l

� �cB cB

B B

/ ² Á %³�²%³�% /² Á %³�²%³�%Á� � ­ ��¦B

so that by (8.40), the dominated convergence theorem, and the Fubini theorem,

� � � �cB cB

B B

¦B$ $

��

/ ²  %³� ² ³ ²%³�% /²  %³� ² ³ ²%³�%À ²�À��³� � � � ­ � � � �

Since is arbitrary, it follows that� � * ² ³�B l

�� � �: ;� cB

%

��

�� � � � � ­ � � �

�� ~ / ²  %³� ² ³ /²  %³� ² ³Â

& b �� �

�

� $ $�¦B

note that the limit is independent of the choice .7�

Similarly, by (8.39),

� � ��

�²�³ ²�³� ~ / ² ÂB³ � ² ³ / ² ÂB³� ² ³Á�

$ $� � � � ­ � � �

�¦B

where , completing the proof of (8.35). / ~ / ʲ�³ ²�³lim�¦B

�

Note that the proof above holds if is replaced by� ��² ³

� � � �� ��

cB

B

�² ³ ~ ²%³� ²%³Á�where is bounded. Thus we have� ��

�²%³ ~ ²%³ b 6²% ³ ²% ¦ �³

Corollary 8.8.1: The statement of the Theorem holds if in is replaced by ²��³ À� �� ��

§8.3 The -Integral9i

Let be a metric space with -finite Borel measure . Let , and² Á ³ ¢ ¦$ � � � � l lb b

�� be the corresponding metric on . Let be -measurable with a range: $ : �? ¢ ¦ Á�

consisting of independent r.v.'s. We now consider the general version of §3.1, i.e., theresult of summing “samples” of at an asymptotically dense set of points in . In? $Chapter 9, will be the spectrum of a von Neumann algebra of physical observables.$

For , let be at most countable. Note the elements of need� $ � $ $� � ~ ¸ ¹ �� �� ��1�

not be distinct. For , let , where denotes cardinality. We. � 5 ².³ ~ O q .O O h O$ $� �

assume that for any open set ²�³ . � Á$

111

� ­ �5 ².³ ².³Á ²�À���³��¦�

and for some , if is an open ball with ²��³ * � ² ³ ) ²)³ � *Á� $ �

� �5 ²)³ � � ²)³ ²�À���³�

for some fixed These conditions should be compared with (3.1).� � l.

Definition 8.9: The net is a - of points. We define¸ ¹$ �� ��� net

9i� �$ �

�?² ³ ²� ³ � ² ³ ?² ³Á ²�À��³� � � � � �lim¦�

�

�

if the right hand limit (in law) is independent of the choice of .¸ ¹$� ���

We proceed to relate the to the Riemann integral.9i

Definition 8.10: Let be a metric space, and . Let diam , and ² Á ³ , � � ~ , � �$ � $be the supremum of the diameters of all balls . The pair are the ) � , ² Á �³ dimensionsof .,

Lemma 8.12.1: Given , there is a partition of such that each has� $� � 7 7 � 7� � ��

dimensions where . Furthermore, can be chosen so that² Á �³ � Á � � 7� � �

�²C7 ³ ~ � ��� for all .

Proof: Let be a maximal set of disjoint -balls in , and be the collection of centers8 � $ *of . For , let be the set of points closer to than to any other˜) � ) ² ³ � )² ³ �8 � 8 � $ ��

point in . Let . If , then the dimensions of satisfy˜ ˜˜ ˜ ˜* ~ ¸)² ³ ¢ � *¹ ) � ² Á �³ )8 � � 8² Á �³ � ²� Á � ³ � � Á � � �� � � �, i.e. . There are at most a countable number of elementsof with non-null boundaries. Let be an enumeration of this collection. For˜ ˜ ˜8 ) Á ) ÁÃ� �

� $ � $ � � � �� �Á . � ) ².³ ~ ¸ � ¢ ² Á .³ � ¹ �, let . There exists such that��

� �

) ²) ³ ~ ¸ ¢ ² Á ³ � � ) ¹��˜ ˜ for some has null boundary, since otherwise � � � � �� � � � � � �

would not be -finite. We replace by and decrease the remaining sets in ˜ ˜ ˜� 8) ) ²) ³� ��

accordingly. We continue in this manner by then replacing by , and,˜ ˜) ) ²) ³Á �� � � ���

��

in general, replacing by , at each stage adjusting as well. At the˜ ˜ ˜) ) ²) ³Á �� � � ���

� �b�� 8

end of this process, all sets have null boundary, and dimensions .˜ ˜) � ² Á �³ � ² Á ³8 � �

We let . ˜7 ~ Ê� 8

Definition 8.11: A Borel measure on a metric space is if for , every ball ofuniform � � �radius has measure bounded below by some constant . A sequence of r.v.'s� � � � ¸? ¹� �

is if the corresponding d.f.'s satisfy , where and are d.f.'s. Itbounded - - � - � . - .� �

is if it is not bounded. A collection of sets is a of a set if itunbounded partial partition $satisfies all the requirements of a partition, except possibly . In particular,�

� �7 ~ $

112

elements of may be apportioned to several partition elements, if non-zero measures are$divided correspondingly.

Note that a sequence of r.v.'s is unbounded if for all there exists ? 4 � � � �� �such that .sup

����7²O? O � 4³ � �

Theorem 8:12. Let be a metric space with uniform -finite Borel measure . If² Á ³$ � � �? ¢ ¦ 9$ : is Riemann-integrable, then it is -integrable, and the two integralsi

coincide.

Proof: Let be an infinitesimal sequence of partitions of , such that (see Lemma¸7 ¹� �Z $

8.12.1) has dimensions dim , and . Since is uniform,7 7 � Á ²C7 ³ ~ �� � �� �� � �Z Z Z� 6 7 � �

there is a function such that . There exists a subpartition of�² ³ � � ²7 ³ � �² ³ 7� � �� ��Z

7�Z , consisting of sets with null boundary, with

�² ³ � ²7 ³ � ��² ³À ²�À��³� � ���

If is non-atomic, can be constructed as a standard subpartition of . If has$ 7 7 7� � ��Z

atoms we can write, according to previous conventions, as a union of copies of7 ���Z

7 ²7 ³ 7� � �� �Z Z�

�, each with measure , with each a distinct element of (this is a quick way�

of eliminating the problem of atoms, although the end result could be accomplished bysubdividing only atoms).

Consider a -net satisfying� $ �� �Z Z

� �~ ¸ ¹

5 ²7 ³ ~ Á ²�À�³� ��Z

��B C��

�

where , and [ ] is the greatest integer function. Given � $ �� �Á 5 ²7 ³ ~ O q 7 O h� �� �Z Z

� �

and , subdivide into the partition , where each element satisfies� 7 ¸7 ¹� �� �� �

� � � � $� � � � ��

��� �� �� �� ��5 ²7 ³Z~ ~ ²7 ³ 7 ��

� �

�Z

�, where , and contains exactly one element .

If , then� �~ �Á ? ~ ?² ³� ��� ��

� � 8@ 9� 8@ 9

�Á� �Á��� �� �� �

c�

�Á�� �

c�

? ² ³ ~ ?

~ ?² ³

� � � �

� �

� � � �

� � �

�

�

�

�

�

�

�

�

AA

²�À�³

where in the last sum is a function of defined by . If increases to� � � 7 ~ ² ³� � � �� �� �

infinity sufficiently slowly as , then by (8.45) and the Riemann integrability of ,� ¦ � ?if ? ~ ?² ³Á� �� ��

113

� :B C ; ��Á�

� ��

c�

¦�? ¬ ? ²� ³À ²�À��³� �

�

� $

� � � ��

� 9

Let denote the greatest integer, be fixed, be an indexing ,´ h µ ¸? ¹ ¸?² ³ ¢ � 7 ¹� � �� � � ��� � � � �

and

� ~ c ² ³À� ��

� ��

c�

� � � ��

�:B C ;To prove that

��Á�

� ��¦�

� ? ¬ �Á ²�À��³� ��

we assume that it is not true. Recall that a space of finite measures with bounded totalmeasure is compact in the topology of weak convergence. Thus there exists a sequence� �M M­

M ¦ B� M ²�³ such that (replacing by )

��Á�

M� M��M¦B

� ? ¬ ²�À� �³�

for some , or � �£ ²��³�

��Á�

M� M��� ? M ¦ B ²�À� �³ is unbounded as .

We assume , since the argument is similar otherwise. By Theorems 7.13, 8.7, and our²�³

hypotheses, is homogeneous of positive order, so that is well-defined, and��

�

c�M�

M

²� ³

vanishes uniformly as .M ¦ BTherefore

B C�

�� ­

��

M

c�M�²� ³ �Á

M ¦ B

uniformly in . Thus we may assume (by taking an -subsequence if necessary) that� M

�B CM

M

M

c�M�

�

�� �

�²� ³ � �² ³Á ²�À�³

for each . For let be a partial partition of , into � M ~ �Á �Áà Á ¸7 ¹ � 7� � ��

��� �M M

� �F > ?��

M

subsets, each of measure , with . By (8.50), can be chosen so� Fc� M MM� M�� �� �²� ³ ? � 7� �

that is a partial partition of . By choosing an -subsequence, we mayF F� ��� �M �M~ 7 M�

assume by that²�³

114

� �3

�Á�

M� M�� M: ;� � ? Á � Á�

*

where may be arbitrarily large. Therefore, the partial Riemann sum * � � � ?��Á�ÁM

M� M��

becomes unbounded, contradicting Riemann integrability of ; this proves (8.48). Thus?by (8.47),

� � � �² ³ ? ¬ ? ²� ³À ²�À�³� ��

�¦�

�� $

9

By an exactly parallel argument, if in (8.45) , and ,� � � � � �~ ² ³ � ~ ² ³ �­ ­� �¦B ¦ �

the latter sufficiently slowly, then

� �² ³ ? ¬ �À ²�À�³��

�¦�

��

Combining (8.51) and (8.52), if , and� �² ³ �­�¦B

B C B C²� c ³ ²� b ³� 5 ²7 ³ � Á ²�À�³

� � � �

� �

� �� �

� �Z�

then

� � � �² ³ ? ¬ ? ²� ³À ²�À�³� ��� $

�¦�

9

Now consider a general -net , not necessarily satisfying (8.53). Fix a� $ �� �~ ¸ ¹� �

partition with null boundaries, satisfying (8.44). If two partition elements7 ~ ¸7 ¹� �� �

overlap (e.g., over atoms), elements of are apportioned among them alternatingly. By$�(8.42),

5 ²7 ³ � O q 7 O �*

� � � �� �$�

for some independent of and , while . Let* � � � 5 ²7 ³� � �� � �� �­

� ¦ �

( ~ � ¢ O 5 ²7 ³ c O ��

�� � � �J K� ��

� �

and

$� � �Á

��(

�~ 7 À ²�À³���

Note that as 0.$ $ �� �Á § ¦We claim that if , then: � ¸� ¢ ¤ ¹� �

� �� $�Á

115

� �² ³ ? ¬ �À ²�À³�:

�¦�

�

��

The proof of this claim, briefly, follows by assuming the negation and forming analternative, as in (8.49). A contradiction then follows along similar lines.

Let , and� � �~ ² ³

$ $iÁ Á ² ³

�

� � � � �

� �

~ À�Z

Z Z

Let ( ) 0 sufficiently slowly that as , and� � $ $ �­

� ¦ 0 iÁ� � § ¦ �

� �² ³ ? ¬ �Á ²�À�³�:

�

�Z

�

if ; this is possible by (8.56). For : � ¸� ¢ ¤ ¹ 7 �� � �� � �Z

� i � i� Á� $ $

O 5 ²7 ³ c O � À�

� ��

� � �� �

Let be constructed so that if , and$ � $ $ $ $� �� � � � �� �� � �

� � � � � iÁ~ ¸ ¹ � q 7 ~ q 7 7 �

$ $� � � ��

� �q 7 q 7differs from by the minimal number of elements such that

e e� ��

5 ²7 ³ c � ²�À�³�

� � ��

� �

for all . Thus by (8.54),�

� � � � �² ³ ?² ³ ¬ ? ²� ³À ²�À ³� ��

��

¦��

� $9

However, is a small perturbation of in that (8.57) and (8.59) imply that (8.59)$ $� �� Á

holds also if is replaced by . � �� ���

� Ê

116

Chapter 9

Joint Distributions and Applications

We now use the mathematical machinery developed in the last two chapters toconsider joint distributions of integrals of r.v.'s and apply them to statistical mechanics.The work in deriving joint distributions of quantum observables will, with the aboveformalism, be minimal.

§9.1 Integrals of Jointly Distributed R.V.'s

As in Chapter 8, let be a metric space with a -finite Borel measure .² Á ³$ � � �

Assume that random vector-valued functions have a jointX² ³ ~ ²? ² ³Áà Á? ² ³³� � �� �

distribution for fixed and are independent for different . Let be� $ � � l l� ¢ ¦b b

given. Define the direct product and for : : � � �� � � � ��~� � � ��~ d ~ ² Á Áà Á ³Á�

� ~ �Á �Áà Á let

� � � �� ��

� �

�~�

�

� �� �² Á ³ ~ ² Á ³� � �

(see equation (7.21)). Define

H � � �ess² ³ ~ ¸ � 9² ³ ¢ ² ²) ² ³³³ � � D � �¹ÀX X X�c�

�

Let ( denote the joint ch.f. of .)X t X³

Theorem 9.1: The random vector exists if and only ifY X~ ² ³ ²� ² ³³$

� � � �

²�³ � ² ³Á for � Hess X

� � ��

X X² ³ � ²" ² ² ³ ³ c �³ ² À�³�

t tlim�¦�

exists, and

117

²��³

� � � �² ³ � ² ³� ² ³ ² À�³t t�$

�X² ³

converges absolutely. In this case, the ch.f. of is Y � À�² ³t

Proof: Suppose exists. If is an -vector and is the log ch.f. of , then forY a t Y� ² ³�

! � ²! ³ hl �, is the log ch.f. of . By Theorem 8.5 the log ch.f. ofa a Ya Y a Xh ~ h ² ³ ²� ² ³³ ²!³ � ² ³

$ $ �� � � � � � � is also given by , adopting the notation ofa Xh ² ³

(9.1) for scalar r.v.'s. For a given , if then � $ H � H� � ²?³Á h ² ³ � ² h ³ÀX a X a Xess ess

Thus, since ) )a X Xh ²!³ ~ ²! ³Áa

lim lim� �¦� ¦�

h� �² ² ² ³!³ c �³ ~ ² ² ² ³! ³ c �³� �) � � ) � �a X X a

exists; since is arbitrary, follows. In addition, by Theorem 8.5,a ²�³

� � � �²! ³ ~ ²!³ � ² ³Áa �$

�a Xh ² ³

and the latter converges absolutely. Using we conclude (9.2)� �a X Xh ²!³ ~ ²! ³aconverges absolutely, and has log ch.f. proving .Y t�² ³Á ²��³

Conversely, assume and hold. Then given and there²�³ ²��³ ? � ² h ³Áa a XZ Hess

exists such that and . Thus by (9.1) andX X X a XZ Z Z Z Z Z� �~ ²? ÁÃ Á? ³ � ² ³ h ~ ?Hess

(9.2) for and then by Theorem 8.5, exists. Since was arbitrary, the proof is�XZ a Y ahcomplete. Ê

§9.2 Abelian -algebras> i

We present here a capsule summary of the spectral theory of -algebras to be used> i

in the next section. The material may be omitted without loss of continuity by thosefamiliar with it.

Let be a separable complex Hilbert space and be the bounded linear> B >² ³operators on . A -algebra on is an algebra of bounded operators on closed> 7 > >> i

under adjunction , and closed in the weak operator topology on . A -²( ¦ ( ³ >i i>algebra is naturally a normed linear space with norm inherited from Let be theB > 7² ³À i

space of bounded linear functionals on as a Banach space and let the spectrum of 7 7:consist of those which are also multiplicative, i.e., , in� 7 � � �� ²( ( ³ ~ ²( ³ ²( ³i

� � � �

the weak-star topology inherited from The Gelfand representation gives a canonical7iÀisometric algebraic star-isomorphism of with the algebra (under multiplication) formed7by the bounded continuous complex-valued functions on . This isomorphism is,* ²:³ :)

for � � Á7

118

� © � � * ²:³Á �² ³ ~ ²�³À) where � �

For let be the corresponding operator. For the map� � * ²:³ ; � %Á & � Á) � 7 >

� ¦ ²; %Á &³ * ²:³� ) is non-negative and is bounded on in its uniform (sup-norm)topology; hence there exists a Borel measure on such that for �%Á& ): � � * ²:³

²; %Á &³ ~ �� À ² À�³� %Á&:

� �

Definition 9.2: The measure is a . A measure on is if for� �%Á& spectral measure basic:any subset of to be locally -null, it is necessary and sufficient that it be locally -: � �%Á%

null for every .% � >

Clearly any two basic measures are absolutely continuous with respect to each other.We have (see [D2])

Proposition 9.3: If is separable, then carries a -finite basic measure.> �:

If is basic, then . If is a measure space, then , as a� � $ $* ²:³ ~ 3 ²:Á ³ 3 ² ³)B B

> 3 ² ³i �-algebra acting on , is the of . We then have (see$ $multiplication algebra[D2]):

Proposition 9.4: Let be basic on . Then the Gelfand isomorphism is the unique� :isometric star-isomorphism of the multiplication algebra onto 3 ²:Á ³ ÀB � 7

If is a -algebra on , a possibly unbounded closed operator is with7 >> (i affiliated7 �7 7 7, or , if commutes with every unitary operator in the commutant of If( ( À (Z

is normal, then if and only if where , and is Borel-( ( ~ �²( ³Á ( � � ¢ ¦�7 7 ] ]� �

measurable.A -algebra is maximal abelian self-adjoint (masa) if it is properly contained in no> i

other abelian -algebra. In physics, a maximal commuting set of observables (that is,> i

its spectral projections) generates a masa algebra. We require:

Theorem 9.5 Two masa algebras are algebraically isomorphic if and only if they[Se1]:are unitarily equivalent.

Definition 9.6: A measure space is if every measurable set is a least upperlocalizablebound of sets with finite measure in the partial order of set inclusion. Two measurespaces have isomorphic measure rings if there exists an algebraic isomorphism betweentheir (Boolean) rings of measurable sets (modulo null sets). They are if theisomorphicisomorphism preserves measure.

Theorem 9.7 Two localizable measure spaces have isomorphic measure rings if[Se7]:and only if their multiplication algebras are algebraically star-isomorphic.

119

§9.3 Computations with Value Functions

Although joint distributions of non-commuting observables in quantum statisticalmechanics are difficult to describe (see [Se6] for the groundwork of such analysis), this isnot the case with commuting observables, which are amenable to a standard(commutative) probabilistic analysis.

Let be an abelian -algebra, and let and . We assume7 �7> ( � � ( ²� � � � M³i� �

that represent globally conserved quantities in a canonical ensemble whose density(�

operator is formally

� ~ ² À�³�

�

c h� ² ³

c h ² ³

� !

� !

A

Atr

where , an -tuple of positive numbers, , and� � �~ ² Áà Á ³ M ~ ²( Áà Á( ³� M � MA� ² ³ ~ ²� ²( ³Áà Á � ²( ³³ ) Áà Á)! ! ! ! ! !A � M ) - � � ( may be either or ; see §1.2). Let be self-adjoint operators affiliated with , whose joint distribution in the canonical7ensemble of (9.4) is to be determined.

As with the single operators, must be interpreted as a limit of density operators with�discrete spectra. Proceeding in an analogous manner, let be the spectrum of . Under$ 7Bose statistics, for , let be the probability space where� $ D t 8 F� ² Á Á ³� �t

bb

t 8 t Fb b c ² ³h ' c ² ³h~ ¸�Á �Á � à ' � Á ²'³ ~ � ²� c � ³, }, is its power set and for ,t �� � � �

bA A

where . Under Fermi statistics,A² ³ ~ ² ²( ³Áà Á ²( ³³� � �� M

D 8 F� �~ ²¸� �¹Á Á ³ÁÁ ¸�Á�¹

where

F F F� � �� �

²�³ ~  ²�³ ~ � c ²�³À ² À³�

� b �c h ² ³A

Thus represents possible particle numbers in state . Spectral multiplicities need notD ��

be indicated here by duplication of spectral values, since they will be subsumed in a

spectral measure . Let be the direct product space with product measure� D D~ ��

�

F F~ ��

�.

Definition 9.8: The pair is the over at generalized² Á ³D F canonical ensemble Ainverse temperature corresponding to the (generally formal) operator .� �

Let be an r.v. on defined by is formally the number of5 5 ² ' ³ ~ '  5� � � � �� $

D �Z

Z

�

particles in state . For let� � � � � �

? ² ³ � ²) ³5� �� � �

be the r.v. on representing the “total amount” of observable in state .D ! �� ²) ³�

120

Our goal is to ascertain the joint distribution of ; this²� ²) ³Áà Á � ²) ³³ � � ² ³! ! !� � Bis the distribution of a formal sum of over . In orderX² ³ ~ ²? ² ³Áà Á? ² ³³ �� � � � $� �

to study the distribution asymptotically, we first center LetX² ³À�

X X X B² ³ � ² ³ c ² ² ³³ � ² ³5 Á ² À³� � ; � � �

where 5 � 5 c ²5 ³Á ~ ²) Áà Á) ³À� � �; B � �

Let be a ( -finite) basic measure (see §9.2) on . We study the Lebesgue integral� � $$ �

� �X² ³²� ² ³³ 5 �� � ��� . In the Bose case the r.v. is geometric with parameter , andc h ² ³A

in the Fermi case it is Bernoulli with parameter given by (9.5). Let be the ch.f. of*�²!³5 ² ³ ² ³� �. If is the ch.f. of , then) �t X

) * �� �² ³ ~ ² h ² ³³À ² À�³t t B

Therefore, for small�

) � � �² ³ ~ � c b6² ³Á�

�t tMt� Z �

where is the covariance matrix of ,4 ² ³ ~ c ²�³�� �CC C

�

! !� �

�� X

4 ² ³ ~ ²) ² ³) ² ³5 ³ ~ ) ² ³) ² ³ ² À�³�

� g ��� � � � �

h ² ³

h ² ³�� ; � � � ��

� �

� �

�

A

A6 7(see equation (2.10)). The holds for Bose (Fermi) statistics. Thus if ,c ² b ³ ² ³ ~� � �

��

� ) � �� ��

² ³ � ² ³ c � ~ c ² ³ À�

�t t tM tlim

¦�

Z6 7��

Together with Theorem 9.1 this proves the reverse direction of

Theorem 9.9: If are the centered random vectors above, then the LebesgueX² ³�integral

Y X~ ² ³²� ² ³³3

���

$

� � � ² À ³

exists if and only if

M M� ² ³� ² ³�$

� � � ² À��³

converges absolutely component-wise. In this case is normal, with covariance matrixY M .

121

Proof: Only the forward direction remains. If (9.9) exists, then by Theorem 9.1,

�$ �

�lim¦�

�² ² ³ c �³ � ² ³

�) � � �

�� t

converges absolutely for all . Thus, the same is true for (9.10). t Ê

§9.4 Joint Asymptotic Distributions

By analogy with single operators, calculations like those above can be viewed as adirect treatment of an infinite volume limit. However, the algebra and its spectrum 7 $are actually limits of a net of discrete algebras with spectra , and the joint7 $� �

distributions of are asymptotic forms of those for discretized random vectors . In theX X �

discretized situation, all formal quantities, including the density operator, are well-defined.

Let be a -finite basic measure on , and be a metric on whose Borel sets � � $ � $ 8are just the -measurable sets (the existence of such metrics in applications will be shown�explicitly; the integral, if it exists, is independent of the metric used). Let be a -net¸ ¹$ ��

(Definition 8.9) of points in , and for , let .$ 7( � ( ~ (�$e

�

Definition 9.10: The algebra is the - of 7 7 � 7� �~ ¸( ¢ ( � ¹ discrete approximationwith respect to . Let and be as in §9.3.� ( ÁÃ Á( ) ÁÃ Á)� M � �

In calculating -integrals of r.v.'s we will be calculating 9 ² ³ ~ ² ³5 ¦ �i X B� � ��

limits of sums

� �� $ � $

� � �

� � � �� �� �

� �X X² ³ � ² ³À ² À��³� �

In order to use the machinery of §8.3, we need some assumptions about and .A B² ³ ² ³� �Precisely, we require that be -continuous, and similarly for (that is, that( ² ³ ) ² ³� �� � �there exist continuous representatives). We then have

Theorem 9.11: Let and be as above, and be uniform (DefinitionA B² ³Á ² ³Á Á Á� � $ � � �8.11) with respect to . If for some � � � �

� 6 7$ � � � �

� �

� �

sup² ³�

�h ² ³

h ² ³�

�Á

O ² ³O � ² ³ � BÁ ² À��³�

� g �B � � �

A

A

then the normalized distribution of in the canonical ensemble over at generalizedB Ainverse temperature is jointly normal, with covariance�

4 ~ ) ² ³) ² ³ � ² ³Á ² À��³�

� g ��� � �

c h ² ³

h ² ³�� 6 7$

� �

� �

� � � �A

A

122

where refers to Bose (Fermi) statistics. That is,c ² b ³

9i�$

B² ³5 ²� ² ³³� � ���°�

exists and is normal with covariance .4��

Proof: By Theorems 8.12, 8.7, and 9.9, we need only show that isX B² ³ ~ ² ³5� � �

Riemann intregrable. By Corollary 8.8.1, it suffices to show

� ² ³ � ² ? ² ³³ b ² ? ² ³³ � 3 ² ³ ² À��³�

� �Á� �

�� � � � �� ; � � � � � � �

�sup� �

� � � �

�

�

�

² Á ³�

6 7l l

for some , and all , where� � � � � � � � �

��²%³ ~% % � �� % � �ÀH

Since ( , letting ; �? ² ³³ � � 0²%³ ~ %Á�

� ² ³ � ? b O²0 c ³² ? ³O�

� � ² ? ³ ~ �) ² ³ ²5 ³�

~ �) ² ³�

�

� �

�

� �

� �

Á ��

�� � � �

�

�� � �

� �

��

h ² ³

h ² ³

� ; � � ��

�; � � L

�

sup

sup

sup

� �

� � � �

� �

� � � �

� � � �

�

�

�

�

�

�

² Á ³�

�

² � ³�

² Á ³�

4 l 7

A

A g �
This shows that (9.14) is implied by (9.12) and completes the proof. Ê

§9.5 Applications

An advantage of Einstein space over canonical spatially cut-off versions ofMinkowski space is its possession of the full conformal group of symmetries. Inparticular, the rotation group acts on Einstein space. Untractable (non trace-class)expressions involving generators of the conformal group in Minkowski space becometractable in Einstein space. For the purpose of evaluating joint distributions ofobservables Minkowski 4-space should be viewed as an infinite volume limit of Einsteinspace.

Theorem 9.11 allows evaluation of joint distributions in canonical ensembles (9.4),viewed as limits (according to a spectral measure ) of systems with discrete spectrum.�This section provides two explicit calculations.

Non-vanishing chemical potential²�³

123

Let be a system with chemical potential , single particle Hamiltonian , andI � � � (formal density operator

� ~ ² À�³�

�

c h� ² ³

c h� ² ³

� !

� !

A

Atr

where , and . Note that is the particle number operator.� � � !~ ² Á ³ ~ ²(Á 0³ 5 ~ � ²0³AThis models an ensemble in which creation of particles requires energy In this case the�À> ( 0i-algebra generated by spectral projections of and is the bounded Borel7functions of .(

If consists of non-relativistic particles in Minkowski -space, the spectrum ofI � b �7 l is measure-theoretically equivalent to . The appropriate spectral measure isb

�� ~ , �, / ~ � ²(³ 5� �c��

!

�°�

�b��6 7 (see (6.12)). The joint distribution of and is!

obtained by integrating , where and is a centeredX B B~ ²,³5 ²,³ ~ ²,Á �³Á 5, ,

geometric r.v. with parameter (under Bose statistics). According to Theorem�c ,c� �

9.11, the normalized joint asymptotic distribution of and is normal with covariance/ 5

M ~ ��À, ,, �

�

²� g �³9i� : ;$

� �

� �

� ,b

,b �

Defining the generalized zeta function

²�Á %³ � � % ² À�³��~�

Bc� �

and using

��

B � %b

%b �c% �

²� g �³�% ~ f �[ ²�Á f � ³ ²� � �³Á ² À��³

�

�

�

we get

M ~ ²� � �³f �[ ²� b �Á f � ³ ²�Á f � ³

²�Á f � ³ ²� c �Á f � ³

�

� !

�� �

� �b��

�²�b�³ c c�

� c c6 7: ;� �� �

�� �

.

Note that and coincides with (6.25) and the right side of (6.21), respectively, in4 4�� ��

the Bose case if . The calculation for is similar, and thus omitted.� ~ � � � �

Density operator involving angular momentum (see [JKS])²��³ In this model, the formal density operator in a system is given byI

� ~  ² À��³�

�

c /c 3

c /c 3

� �

� �

�

�tr

124

where and are energy and angular momentum [JKS, Se8]; we ignore chemical/ 3�

potential for simplicity. We will study in Minkowski four-space , as an infiniteI 4 �

volume limit of Einstein space.We remark first on the appropriate measure space of Theorem 9.11. Let be the7i

> ( �i �-algebra generated by and , the single particle energy and angular momentumoperators. Let be the spectrum of and be a -finite basic measure on By$ 7 � � $i i i iÁ ÀProposition 9.4, is star-isomorphic to 3 ² ³ ÀB i i� 7

Let , and , with Lebesgue measure, and ,$ l t �~ d ~ �d � � �¸�¹ ~ �� b �b b

for . Then is star-isomorphic to , specifically through the� � ~ 3 ² ³t 7 � 7b B i

correspondence

( © 4  � © 4 ² À� ³,�

�²�b�³

where and denote independent variables on and , respectively, and denotes, � 4l tb b

multiplication. Thus and are star-isomorphic. By Theorem 9.7, therefore,3 ² ³ 3 ² ³B i B� �² Á ³ ² Á ³$ � $ �* i and have isomorphic measure rings, so that the image (under theisomorphism) of on is equivalent to , and thus itself a basic measure.� $ �i i

Thus, henceforth we may restrict attention to is a physically appropriate² Á ³Â$ � �measure on , since it incorporates the asymptotics of the joint spectrum of and in$ ( �� �

�

Einstein space of radius proportional to as becomes infinite. Specifically< 9 Á 9� ��

(see §6.3), the joint spectrum of and in is( � <� �� �

¸² �Á �²� b �³³ ¢ �Á � � ¹Á� tb

this being the joint range of the corresponding functions on .$The object of interest in studying the asymptotics of the joint distribution� ¦ �

(covariance) of and is, according to Theorem 9.11, the local covariance of / 3 ,5��

and , where , and is a centered geometric r.v. with�²� b �³5 ~ ²,Á �³ � 5� �� $

parameter , with and . This is the dyadic matrix� ~ ² Á ³ ² ³ ~ ²,Á �²� b �³³c h ² ³� �A � � � �A

M² ³ ~ ², �²� b �³³ ²5 ³ ~ À, , ,�²� b �³

�²� b �³ ,�²� b �³ � ²� b �³

�

� g �³� L: ; : ;�

�

�

�

� �

h

h �

A

A(

Integrating over , we obtain as the covariance of and in Minkowski space$ / 3�

M ~ ²�Á ³Á ² À��³� c

c�� �: ;

� � CC

� C CC C

g� � �

� � �

�

�

�

where

� � g

�~�

Bc �²�b�³²�Á ³ ~ f ²�� b �³ ²�Á f � ³ ² À��³� �

with given by (9.16). Note that, as indicated by the limit in (9.21), the joint � ¦ �distribution of energy and angular momentum when is singular, with the� ~ �conditional expectation of angular momentum infinite for every energy value in

125

Minkowski space. This is to be expected, since for a fixed energy value the range of theangular momentum becomes unbounded as . See [JKS, Se8] for an application of9 ¦ B(9.20) to a model for the influence of angular momentum on the cosmic backgroundradiation.

126

Epilogue

I would like to leave the reader by briefly identifying two significant open questionswhich arise in the present context.

The first is, what asymptotic probability distributions arise in a system whosespectral measure fails to have three continuous derivatives near 0� � % �% ²� � �³� �

(Definition 3.11)? The failure to answer this question in Chapters 4 and 5 seemstechnical.

Second, how can these results be extended to a non-commutative setting (i.e., oneinvolving non-commuting observables)? It seems that gage spaces, the non-commutativeanalogs of probability spaces [Se6], are the appropriate framework. This question mayhave very significant mathematical ramifications.

127

References

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[BD] Bjorken, J.D. and S.D. Drell, , McGraw-Hill,Relativistic Quantum Fields New York, 1965À

[BDK] Bretagnolle, J., D. Dacunha Castelle, and J. Krivine, Lois stables et espaces , 2 (1966), 231-259.3� Ann. Inst. H. Poincaré

[Ch] Chung, K.L., Academic Press, New York,A Course in Probability Theory, 1974.

[Che] Chentsov, N.N., Lévy-type Brownian motion for several parameters and generalized white noise, 2 (1957).Theor. Probability Appl.

[D1] Dixmier, J., - , North Holland, Amsterdam, 1977.*i Algebras

[D2] Dixmier, J., North Holland, Amsterdam, 1981.Von Neumann Algebras,

[GS] Gelfand, I.M. and I.S. Sargsjan, , A.M.S.,Introduction to Spectral Theory Providence, 1975.

[GV] Gelfand, I.M. and N.J. Vilenkin, , vol. 4. Generalized Functions Some Applications of Harmonic Analysis, Academic Press, New York, 1964.

[GV] Gnedenko, B.V. and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, 1968.

[GR] Gradshteyn, I.S. and M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965.

[G] Gumbel, E.J., Columbia University Press, New York,Statistics of Extremes, 1958.

[H] Hörmander, L., The spectral function of an elliptic operator, 121Acta Math. (1968), 193-218.

[KL] Khinchin, A.Y., and P. Lévy, Sur les lois stables, C.R. Acad. Sci. Paris 202 (1936), 374-376.

[JKS] Jakobsen, H., M. Kon, and I.E. Segal, Angular momentum of the cosmic background radiation, 42 (1979), 1788-1791.Phys. Rev. Letters

128

[Kh] Khinchin, A.Y., GraylockMathematical Foundations of Quantum Statistics, Press, Albany, N.Y., 1960.

[La] Lamperti, J., Semi-stable stochastic processes, Trans. Am. Math. Soc. 104 (1962), 62-78.

[LL] Landau, L.D. and E.M. Lifschitz, , Addison-Wesley, Reading,Statistical Physics Mass., 1958.

[Lé] Lévy, P., A special problem of Brownian motion and a general theory of Gaussian random functions, in Proceedings of the Third Berkeley Symposium on Mathematical and Statistical Probability, 1956.

[Lo] Loève, M., Springer-Verlag, New York, 1977.Probability Theory, I,

[Mc] McKean Jr., H.P., Brownian motion with several dimensional time, Teor. Verojatnost. i Primenen 8 (1963), 357-378.

[M] Malchan, G.M., Characterization of Gaussian fields with the Markov property, Soviet Math. Dokl. 12 (1971).

[R1] Rozanov, Ju. A., , Springer-Verlag, New York, 1982.Markov Random Fields

[R2] Rozanov, Ju. A., On Markovian fields and stochastic equations, Mat. Sb. 106(148) (1978), 106-116.

[See] Seeley, R., An estimate near the boundary for the spectral function of the Laplace operator, 102 (1980), 869-902.Am. J. Math.

[Se1] Segal, I.E., Decompositions of Operator Algebras: II: Multiplicity Theory, American Mathematical Society Memoirs 9 (1951), 1-66.

[Se2] Segal, I.E., AcademicMathematical Cosmology and Extragalactic Astronomy, Press, New York, 1976.

[Se3] Segal, I.E., Tensor algebras over Hilbert spaces. I., 81Trans. Am. Math. Soc. (1956), 106-134.

[Se4] Segal, I.E., Tensor algebras over Hilbert spaces. II. 2Ann. Math. (1956), 60-175.

[Se5] Segal, I.E. and R.A. Kunze, , McGraw-Hill, New York,Integrals and Operators 1968.[Se6] Segal, I.E., A noncommutative extension of abstract integration, 57Ann. Math.

129

(1953), 401-457.

[Se7] Segal, I.E., Equivalences of measure spaces, 73 (1951),Amer. J. Math. 275-313.

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[Si] Simon, B., Schrödinger semigroups, 7 (1982), 445-526.Bull. Am. Math. Soc.

[V] Vanmarcke, E., R , M.I.T.andom Fields: Analysis and Synthesis Press, Cambridge, U.S., 1983.

[WW] Whittaker, E.T. and G.N. Watson, CambridgeA Course of Modern Analysis, University Press, London, 1973.

130

Index

almost surely (a.s.), 6, 38angular momentum, 1, 4, 123-125 covariance with energy, 124antisymmetric distributions, 32 occuptation numbers, 32 statistics, 8, 32antisymmetrized tensor product, 8asymptotic distributions, under symmetric statistics, 27, 47, 64, 65-68, 71-74 under antisymmetric statistics 32, 33asymptotic energy density, 67, 68, 70, 74asymptotic energy distributions, 67asymptotic probability distributions, 126asymptotic spectrum, 15, 16Banach space, 117basic measure, 118, 124Bernoulli, 14, 32, 120Bessel function, modified, 64blackbody curve (classical), 60Boltzmann's constant, 4, 13Borel measures, 83 (finite), 86, 95-6, 111 (metric space), 118 (measurable) positive, 86 set 83, 94, 121 Borel measurable, 96�� c -finite, 110, 116�Bose-Fermi ensembles, 6, 74 statistics 74, 119, 120, 122Bose condensation, 74Boson field, 7 energy, 70 number, 72-3bounded functions, 123bounded r.v.-valued function, 85, 95bounded variation, 18, 27, 28Brownian motion, 76Cauchy data, 63, 64characteristic function (ch.f.), 6, 54, 60, 61 , 106, 116Á �canonical ensemble (over an operator), 11, 13, 17, 119, 121central limit theorem, 23chemical potential, 6, 60, 123, 124 non-zero, 123

131

non-vanishing, 122convergence in law, 6, 24, 42, 86convergence distributional, 38 order-independently, 78, 87 weakly, 77-8, 80-1, 100, 101convolution, iterated, 23, 53 (exponential)countably additive measure-valued measure, 103distribution function (d.f.), 6, 78 (cumulative)density of states, 60, 61density operator, 4, 11, 64, 119, 123density (mean) of photons, 74dimensions (of a set), 111dimensions, physical, 6Dirac delta distribution, 63discretized Gaussian, 16dominated convergence theorem, 110dyadic matrix, 124eigenvalue density, 17Einstein space, 4, 5, 60, 69, 71, 75, 122, 124 distributions in 2 dimensions, 124 distributions in 4 dimensions, 70-75 comparison with Minkowski space, 122elliptic operator, 4, 16 (differential)energy density spectrum, 5energy distribution function, 67�-discrete approximation, 121�-discrete operator, 17, 18, 29, 61, 64, 71�-net of functions, 17equilibrium quantum system, 4Euler's constant, 47even partition, 21, 22extreme value distribution, 47, 60Fermion field, 7 number 68, 69 (ensemble), 73, 74Fourier transform, 51, 63Fubini theorem, 110Gelfand isomorphism, 118Gelfand representation, 117Hamiltonian, 4, 5, 9, 11, 13, 20, 60, 64, 69, 70, 74, 123Hilbert space, 7, 9, 39, 62, 70 complex, 117Huygens' principle, 69hyperbolic equation, 63infinite volume limit, 60

132

infinitively divisible distributions, 94infinitesimal array of r.v.'s, 77 net of partitions, 85 sequence of partitions, 86infrared catastrophe, 34, 74integral of r.v.'s, 20, 76, 94, 116 of measure-valued functions, 71, 83integration by parts, 101integration operator, 94inverse temperature, 4joint characteristic function, 14joint distributions, 119-122, 123, 124joint spectrum, 124Khinchin, A.Y., 5Laplace-Beltrami operator, 69Lebesgue integral, 5, 7, 76, 83-89, 90 (def.), 91, 120 additivity, 87 existence, 96, 100, 103, 104, 120 finite, 85, 86 integrable, 108 integration, 76 general, 85, 102 measure, 124 of r.v.-valued function, 95 of vector functions, 116 -Lebesgue integrability, 95� relation to Riemann integral, 106, 107 sums, 87Lévy-Cramér convergence theorem, 37, 51Lévy-Khinchin transform, 77, 78, 80, 96, 104Liapounov's theorem, 24, 25limit theorems (general), 77limiting distribution, 23localizable (measure space), 118Loève, M., 77, 78, 105log ch.f., 94Lorentz, 63many particle state, 11maximal abelian self-adjoint (masa), 118mean value theorem, 79, 82, 109measure ring isomorphisms, 118measure space, 118metric Lévy, 83

133

, 84��metric space, 111monotone decreasing, 36 non-decreasing, 41, 43multidimensional white noise, 5multiplication algebra, 118Minkowski space, 4, 5, 60, 62, 67-9 ( -dimensional), 74 ( space)� b � � b � 75, 122, 123 ( space), 124, 125� b �net of operators, 4, 61 (discrete)non-atomic, 85, 112non-commutative probabilities, 126non-degenerate interval, 21non-increasing function, 18non-negative, 6, 10non-normal distributions, 6non-vanishing densities near zero, 51number random variable, 14,15occupation number, 13, 14, 29, 41 (total), 55, 61one parameter family of r.v.'s, 20order-independent, 79orthonormal eigenbasis, 13 basis, 9, 10�-bounded, 85particle number operator, 123partition, 20, 21 disjointly, 86 even, 21photons and neutrinos, 60, 74-5Planck law, 4, 5, 6, 60, 74 constant), 75point mass, 87positive, 6, 20 (numbers)positive reals, 93probability distributions (of observables), 1, 23, 87-8, 94pure point spectrum, 11, 13quantization (of an operator), 99iintegral, 110 definition, 110, 111 relation to Riemann integral, 111random distributions, 5random variable (r.v.), 6, 23, 34, 47 (geometric), 64random variable on canonical ensemble, 14random variable-valued functions, 5, 6, 94-5, 103, 106random variables with infinite variance (sums of), 77random vectors, 116range, essential, 85

134

Riemann integral, 5, 20 (Riemann-Stieltjes), 76, 106, 107 ( -Riemann)� definition, 106, 107 geometries, 60 integration, 76 integrable, 108, 112, 114, 122 manifolds, 65, 76 partial Riemann sum, 114 relation to Lebesgue integral, 94, 107 relation to -integral, 1119i

zeta function, 65Riemannian manifold, 4, 16scaling stable, 88, 89, 91stable (scaling) distribution, 88-90Schrödinger operator, 60, 61Segal, I.E., 4, 6self-adjoint, 9, 10self-adjoint operator, 16semi-stable process, 91semi-stable stationary 0-mean, 92separable, - , 95�sequence of partitions, 22�-finite basic measure, 124 measure space, 85single particle energy, 124single particle space, construction, 63-65single particle state, 11singletons, 89singular integrals, 26, 34-36spectral approximation, 61, 64, 65spectral density (non-vanishing) 5, 16spectral function, 17, 29, 38, 43, 57, 61spectral measure, 4, 17, 18, 19, 20, 27, 39, 47 (with non-vanishing density), 60, 61, 64, 65, 71, 118, 123, 126spectral -net, 20, 21, 23, 25, 27�stable distribution, 23, 24, 88 characterization, 88, 89 star -isomorphic, 124standard deviation, 23state probability space, 10, 11Stieltjes integral, 17, 20, 31Sturm-Liouville systems, 16symmetric canonic ensemble, 11, 29symmetric occupation number r.v.'s, 39symmetrized tensor product, 7tail behavior of probability distributions, 83

135

tails of integrated distributions, 77topology of weak convergence, 77trace class, 4, 10, 11, 12, 18, 19 under antisymmetric statistics, 11uniformly bounded, 82value function, 5, 6, 10, 11, 13, 29von Neumann algebra, 76 (abelian), 110> i-algebras (abelian), 117 definitions, 117, 124 properties, 117, 118without loss of generality (w.l.o.g.), 7, 34, 81, 87, 92, 990-free (operator), 11, 19, 270-freedom condition, 320-free spectral measure, 190-mean random variables, 23

136

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